The title says it all.
A character falls off or gets knocked off a flying mount.
Item kicks in to feather fall him.
Does the character at that point have enough time to cast or activate an item that can fly?
It depends on how high you are when you start falling. You can't cast anything when the item kicks in, but you can if you are still falling when it comes to your next turn.
As long as you are higher than 60' you may be ok. 120'+ might be necessary depending on whether your GM has you fall at the initiative count you are knocked off, the beginning of your turn or the end of it. I'm not sure the rules specify when you fall in slow-mo. I'd probably go with the beginning of your turn in which case you need to be higher than 120' (60' straight away, 60' at the beginning of your turn).
From my understanding you would usually be knocked from your mount outside your own turn and start falling 60' a round. I don't think it is specified when exactly the falling part happens but you get to act within that initial fall as a complete round always includes your own turn.
The worst case scenario would be getting knocked off your mount as a reaction to something you do (during your turn) without having any action left that allows you to fly. The longest fall that could be argued to be within the rules in that case would be 5 (ring activates) + 60 (first round) + 60 (second round before you get to act) = 125 feet. But at most tables it would be probably more like 60 (first round) + 30 (second round before you get to finish casting fly halfway through) = 90 feet.
You will probably have to make a pretty tame concentration check (DC 10 + spell level for vigorous motion) to cast the spell itself while falling at the decreased speed.
A character cannot cast a spell while falling, unless the fall is greater than 500 feet or the spell is an immediate action, such as feather fall.
Feather fall instantly changes the rate at which the targets fall to a mere 60 feet per roundCasting a spell while falling requires a concentration check with a DC equal to 20 + the spell’s level.
Feather Fall doesn't mention if this Concentration check goes away or not with the reduced falling speed, though you're still technically falling, so RAW, the full Concentration check is still in place. I can't find anywhere that states how far a creature falls per round, so it's unclear if the 500' referenced is supposed to be all at once, or divided up into two rounds - like Dragonhunterq said, this depends largely on at what point in your character's turn does the falling distance occur. However, I DID find this:
Characters on a plane with subjective directional gravity can move normally along a solid surface by imagining “down” near their feet. If suspended in midair, a character “flies” by merely choosing a “down” direction and “falling” that way. Under such a procedure, an individual “falls” 150 feet in the first round and 300 feet in each succeeding round.
This adds up to 450', NOT 500', so it's not a perfectly lateral rules interpretation, but it's close enough to lead me to believe that if while falling normally, you can't cast any spells unless the fall is greater than 500', and it takes 2 rounds of "falling" under the Subjective Directional Gravity rules to go 450', that the movement occurs at the beginning of your turn, before you take any actions, which by extension, would mean the same thing for Feather Fall.
Going outside the rules for a moment, using normal Earth gravity (accelerating at 32'/second), in a 6 second round, you'd fall a total of 672', so assuming Pathfinder gravity is supposed to be comparable to earth's gravity, neither the standard falling rules nor the Subjective Directional Gravity rules come really close to representing a realistic fall time.
Not sure if I've answered the op's question or not; my main goal was to just lay everything out - falling is quite a mess of rules and assumptions in pathfinder. My personal rules interpretation is that you'd need a total of 120' to fall under the effects of Feather Fall before being able to cast a spell (requiring a Concentration check of 20 + spell level, though maybe a nice GM should lower that to 15 + spell level - falling 10'/second with nothing under your feet would still be quite distracting, I'd imagine). That's just my call after filling in some blanks with my own reasoning, however.
I'm pretty sure you are normally assumed to fall 500 foot during the first round, if only because they would tell us if falling caused a spellcaster to skip an entire turn. That distance is also relatively close to the real world physics you described and in line with the rules for inversed gravity. Subjective directional gravity seems to be an (understandable) outlier.
A severe fluctuation sends the creature falling upward for 2d6 rounds, for a distance of 500 feet in the first round and 1,000 feet in each successive round.
So the falling passage is likely supposed to tell us that a character has to fall first in a round before he can attempt to cast. If your character possesses a ring of feather falling this means he would get his first opportunity after falling 60 feet of distance (potentially more if he started falling during his own turn with no standard action left).
Additionally I'm pretty sure the DC 20 + spell level is specifically for free falling, you are totally right that RAW it doesn't explicitely say so.
Then again I think that casting while free falling at 500-1000 feet a round should be harder than doing so while falling at a regular persons jogging speed and so I would houserule it without a second thought. Personally I'd simply use the DC of 10 + spell level for vigorous motion (like casting from a horse moving 100' a round) I mentioned before. We have to keep in mind the caster is trying to cast fly, so he is probably used to having nothing under his feet by now. The somewhat difficult part is the uncontrolled (but predictable) movement, which is similar enough between a fall and a horse ride that the difference in speed (the horse being almost twice as fast) evens it out.
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The question I think that needs to be asked is... do you want a caster to be able to cast while falling? Because if you put the height above 100' it's not likely they will ever get the chance too. But if the height is 60' then it will happen often enough to matter.
I think it would be better to allow the character a chance to recover before continuing the fall on their turn.
I understand the real world physics of falling. However the reason for the question, is if there is enough time for a cloak which takes a standard action to activate, will be able to be used before impact.
If featherfall were a prepared spell, it can be cast as an immediate action.
This means it can be cast out of turn (but takes up your swift action for next turn) unless you already used an immediate action.
If the item behaves like that, the character simply falls 60' per round. So on the round the character wants to cast Fly, the spell comes into effect after falling 60'... of however high they were when they were dismounted.