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.
