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I effect they operate as a kind of hybrid creature and your fine to use a bit of logic in your games as a GM to make sure that your characters are being properly challenged. Something to just help out in general is not to forget that cat might be entering some heavier encumbrance level due to armored cavalier riding it about. All that gear and the person add up to that cat not being very agile anymore.
Some suggestions for your specific questions:
You should be able to ride a swimming mount though I suggest a penalty due to the added bulk of something on its back. -1 or -2 should be fine. The mount makes the swim check. This is of course assuming the mount is swimming along the surface of the water. Not exactly sure how you'd have it swim downward, probably something in Handle Animal that would work or just assume it doesn't know the appropriate trick and make the DC 25.
If they fall together it is pretty bad for them. Both are going to take falling damage and likely at least the mount could take some extra damage based upon the size of the rider using the Falling Objects table in the Environment chapter of the core book.
What happens when he swims while on the mount - does he have to make ride checks while the mount makes swim checks? If he falls into water does he fall off the mount?
Both actually, He has to make a DC5 ride check to stay in the saddle, and the mount has to make appropriate swim checks. If he has a combat trained mount and the mounted combat feat, there's little reason to bother rolling this at all.
If he jumps or falls a long distance do both he and the mount take damage? If the mount makes an acrobatic check does he also have to make one - or does he make a ride check?
They are both falling, he's just stopping against the mount instead of the ground. Just because what he's landing on is alive doesn't mean he doesn't suddenly and violently decelerate against it. That in mind, they should both roll falling damage and he should make an appropriate ride check to stay mounted.
If the mount makes an acrobatic check does he also have to make one - or does he make a ride check?
Ride check again. DC5 to stay in the saddle, so it's generally not worth rolling unless they haven't bothered to put points in the skill at all.
This makes riding the mount really easy for cavaliers, paladins, samurai and other classes with access to combat trained mounts. I believe that this is as intended, because with an increased risk of being dismounted, you'd have more cases of these classes and builds falling behind other builds dramatically as they would occasionally fall off the mount and be prone, seemingly at random.
Don't forget Ride is affected by Armor Check Penalty of the heavily armored rider. If the cat is doing things like climbing up trees and doing backwards flips you would be warranted in adding and additional -2 (or more) penalty to that check.
Also, I'd count cats as ill-suited as mounts "If you attempt to ride a creature that is ill suited as a mount, you take a –5 penalty on your Ride checks." If cat is not combat trained rider must use move action to make a DC20 check every round to control it.
I disagree with you on these two points in particular, and only because it was specifically stated that it was a cavalier in question. The rules as written for cavaliers overwrite the normal penalties for mounted combat, but only with regards to the cavalier's chosen class mount.
A cavalier does not take an armor check penalty on Ride checks while riding his mount. The mount is always considered combat trained and begins play with Light Armor Proficiency as a bonus feat. A cavalier's mount does not gain the share spells special ability.
A cavalier's bond with his mount is strong, with the pair learning to anticipate each other's moods and moves. Should a cavalier's mount die, the cavalier may find another mount to serve him after 1 week of mourning. This new mount does not gain the link, evasion, devotion, or improved evasion special abilities until the next time the cavalier gains a level.
If the cavalier is riding a cat, it is likely that he is a beast rider cavalier, in which case he would be allowed to ride a cat with no penalty, as shown below.
Medium beast riders can choose a camel or horse mount at 1st level. At 4th level, a Medium beast rider can also choose an allosaurus, ankylosaurus, arsinoitherium, aurochs, bison, brachiosaurus, elephant, glyptodon, hippopotamus, lion, mastodon, megaloceros, snapping turtle (giant), tiger, triceratops, or tyrannosaurus as his mount. Additional mounts might be available with GM approval.
The obvious downside being that if he chose a lion or a tiger, they would be large and therefore unable to fit through doorways reliably. Now, it wasn't specifically stated that he was a beast rider. If he's not a beast rider, you are absolutely correct and those penalties need to be applied, because it's not his class mount and is therefore unsuitable and probably not combat trained.
I think that most quadrupeds can be excluded from the "ill-suited mount" penalty if they're trained for riding. Granted, a tiger or zebra might not have properly shaped withers for a riding saddle, but I don't think the game is supposed to take such real world concerns into account when somebody wants to saddle up Battle Cat and ride off to adventure.
That said, the DM should feel free to call for Ride checks to stay in the saddle whenever it seems reasonable. If you max out your Ride skill it should be pretty easy to make the checks, but less skilled riders might have some comic/tragic misphaps.
Swim is really just an issue of weight and encumbrance. If the mount is armored or carrying a medium load, he'll take penalties on the swim check.
In theory, Climb should be trickier because the rider's weight placement works against the climber. Looking at the Climb rules, though, there's not any adjustment for weight placement (i.e., a backpack vs. armor), so RAW, that's not an issue. In a home game, I would probably add 5 to the DC for the cat's climb which could be negated by a Ride check (representing the rider leaning forward or otherwise adjusting his weight to help the mount climb).
Jump would be covered by the Leap rules under the Ride skill description (http://paizo.com/pathfinderRPG/prd/skills/ride.html#_ride). Also note the rules for attacking from the back of a mount there, also.
If I were GMing, I would take some time to discuss the Ride skill, Handle Animal skill, and the various Mounted Combat feats with my player in advance. If we start out in agreement on these, it would really speed up game time.
You might also want to add a summary of the Ride and Handle animal check DCs to your GM screen or reference material. When I'm playing my cavalier in PFS, I keep a cheat sheet of the Ride and Handle animal check DCs in my character folder, along with a copy for the GM. It really, really helps!
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There are a lot of benefits to being mounted, however avoiding fall damage is not one of them.
Here's a resource for resolving wacky problems with mounted characters. It even has riding cats - and uses for the Ride skill which might cover your falling cat.
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Specific answers to the questions you pose:
(I will assume an appropriately-sized character riding an appropriately-sized mount. I will also assume the GM allowed a cat to serve as a mount since the cat is not a choice listed under the cavalier class.)
1- To swim across water while mounted, the mount makes a Swim check (DC10/15/20, modified by encumbrance) for each round it is in the water to move at 1/4 speed as a move action or 1/2 speed as a full round action. If the mount goes underwater (fails by 5 or more) then I would require a Ride check to "stay in saddle" (DC5), perhaps modified by how many points the check was missed. You might also require a Handle Animal check to "push" a cat to willingly swim across water (DC25) - this is a cat we are talking about.
2- If a mounted rider were falling (a non-deliberate fall, into a pit perhaps), then apply damage to the cat per the falling rules in the CRB (1d6/10' fallen). Since lethal damage will be taken by the mount, then the mount ends up prone. Since the mount has fallen, then (according to the mounted combat rules in the CRB) the rider must make a Ride check to make a "soft fall" from the mount(DC15) or the rider takes 1d6 damage and is also prone. You may want to modify Ride check as if it were a "leap" of the distance fallen. Regardless, this is generous to the rider and uses the mount as a cushion, but I think it is within the rules.
For a deliberate fall ("jump down to that ledge, kitty"), you should require a Ride check for the rider to both iniitate the action and also to remain seated (see the "leap" action under the Ride skill and modify appropriately for the height of the jump). Also, have the cat make an Acrobatics check (DC15) to mitigate damage from the fall as described under the falling rules. If the cat takes any lethal damage from the fall then it ends up prone, and apply the same consequences to the rider as described above for a non-deliberate fall (i.e. a Ride check for a "soft fall") .
3- If the mount is making Acrobatics checks at the direction of the cavalier, then this would either require a Ride check (to intiate a "leap") or possbly a Handle Animal check (to handle or "push" the mount to do something else, like walk along a narrow ledge or across a slippery surface). Also, remember that according to the combat while mounted rules, if the mount moves more than 5' in a round then the rider can only make a single melee attack at the end of the mount's movement; missile attacks take a -4 penalty is the mount double moves and are also made at the end of the mount's movement; and spellcasting while the mount is moving may require a concentration check.
Overall, I suggest that you and your player take a close look at the appropriate rules for the Ride skill, Handle Animal skill, rules regarding mounts as animal companions (for the Cavalier in this case) and examine the mount closely for issues regarding tricks, combat training and how encumbrance might effect the cat's skill checks. There are also many feats that can modify mounted combat.
Some further details to consider:
First, riding the cat requires an exotic riding saddle or an exotic military saddle. If riding bareback there is a -5 penalty imposed on on all Ride checks made by the rider. Regardless, the Cavalier still needs to make the appropriate Ride checks but, as a Cavalier on his mount, many of these will be easier because the Cavalier does not take an Armor Check Penalty. However, he still needs to make the checks.
Some of these may be automatic (assuming the Cavalier has at least one rank in Ride), such as "guide with knees" (DC5) and "stay in saddle" (DC5). Even "fight with a combat-trained mount" (DC10) should be easy to accomplish with only a few ranks and/or a positive Dex modifier. Relevant checks that might not be automatic until the Cavalier gains several ranks in Ride include "cover" (DC15) "leap" (DC15) and "spur mount" (DC15) Note that some of these Ride skill checks may require an action from the rider and, if even if successful, the mount may need to make an appropriate skill check (such as Acrobatics check to jump after the "leap" check).
As for the cat's actions, these can be directed by the Cavalier, whether he is riding or not, with a Handle Animal check. I suggest that the cavalier invest in at least one rank of Handle Animal skill: it cannot be used untrained and they also get a +4 bonus to handle their mount, so getting the animal to perform tricks it knows will be easy, "pushing" untrained tricks is easier and the cavalier can teach them new tricks too.
On the part of the cavalier, getting animal to perform a trick it knows (like "attack") is a free action (DC10, or DC12 if the animal is wounded) and "pushing" it to perform a trick it does not know (like "fetch", "flank" or "track" for example) is a move action (DC25, or DC27 if wounded).
The big cat begins with Int 2, giving it seven tricks known (six for Int and one bonus for being an animal companion). As per the Cavalier class, the mount is "combat-trained" but combat training encompasses six tricks (attack, come, defend, down, guard, and heel), leaving only trick available level one (this should be wisely spent on another "attack" trick to the cat would not need to be "pushed" to attack undead or abberations). As mentioned above, untrained tricks require thatthe animal be "pushed, which may be difficult for the cavlier to accomplish, and also require a move action.
Finally, just like for characters, the greater of the penalties from armor or encumbrance (if any) should also be applied to the base move, ability checks and skill checks made by the cat. If the cat wears armor/barding (ss per the Cavalier class, the mount begins with the Light Armor Proficiency feat) then there may be an associated armor check penalty to Str/Dex skills (like Climb, Swim and Acrobatics) that should be applied to the mount's skill checks. In addition, encumbrance for the cat should be calculated to include the weight of the gear worn or carried by cat (such as the saddle, saddlebags and their contents), weight of the armor/barding worn and the weight of the rider and his gear (this includes the weight of his armor and weapons).
A size M big cat mount starts with a Str 13, so even as a quadruped (carrying capacity 75#/150#/225#) it will likely be penalized with medium encumbrance if carrying a halfling in full plate with a heavy shield, sevaral weapons, and adventuring gear. Medium encumbrance reduces the cat's base speed from 40 to 30 and imposes a -3 to Str/Dex skill and ability checks. However, when the cat reaches size L at level 7 it also gains a +8 Str bonus, giving it least a Str 21 (light load for a large quadruped <= 612#) and allowing it to easily carry even a heavily-armored, fully-equipped half-orc.
By the way, I recently reviewed the Ride and Handle Animal skills and the relevant rules regarding mounted combat and animal companions for two characters I was building (a humam cavalier with a horse and a halfling druid with a big cat) - if I have made some errors in the above, I would appreciate someone pointing them out to me.