Yes.
There are no limits* to how high a pixie may fly, provided that they end their turn only one square above the ground. However, if they do end their turn above their altitude limit they fall with the consequences laid out for falling (1d10 damage/10 foot of drop).
*The theoretical maximum height a pixie may reach on their turn (with no extra movement trickery) is 8*3 + 1 or 25 squares high (run 8, trade standard for another run 8, action point for another run 8, +1 for starting height), that's 125 ft at the end of which they would fall. Falling says to subtract the creature's fly speed so that leaves us at 95 feet of fall, meaning 9d10 falling damage.
What Happens if the Rider undergoes forced movement?
PHB pg 198 also has a partially relevant sentence:
If you’re knocked prone while mounted, you must make the same saving throw.
While none of your above examples involve the 'Prone' status affliction, we can reason that under normal circumstances, if something were about to try and force move you (effectively off your mount), then you would be subject to the same saving throw to remain mounted as outlined above as you would if the mount were also being forcefully moved. (DC 10 Dex).
What about the military saddle?!
If you add in the military saddle, then you would have advantage on the DC 10 Dexterity saving throw to remain mounted.
This is an example of 'Specific beats general' rule of thumb. Were the general rule is found in spells like Thunderwave, that do not allow you save, they just work, and push you up to 15 feet; and the specific rule is what happens when you are mounted and you (or your mount) are forcefully moved while using a military saddle.
What about being grappled in your military saddle?
The grapple rules are on PHB pg 195:
"...you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you succeed, you subject the target to the grappled condition..."
Grappled: Your speed becomes 0...
Where as the other examples are doing something to impact you; a shove, a shockwave, a hard attack, the person grappling you is physically picking you up and moving you against your will (even though your speed is 0). So I would personally rule that you gain advantage on the check (because of the saddle) to avoid the grapple condition but once you fail and become grappled, you are at the mercy of your captor. You can continue to save and break free of the grapple, but you get no more advantage, and can do nothing to control being forcefully moved out of the saddle as you are grappled and no longer able to hold on to anything.
Application of the above example: Let's say a Kraken tried to grappled you (shame on you for riding your horse on the ship deck!), you could try and hold on to the saddle and fight away the tentacles, but if they overcame your resistance and wrapped themselves around you; you would be effectively helpless to do anything that would gain you advantage once it tried to pull you out of the saddle and see how tasty you are. You struggle as you are lifted out of your mount and over the water, drawing closer to the creatures maw, but you break free on your own just before you become Kraken food.
Best Answer
There are no rules that allow for this.
The Player's Handbook/basic rules for falling provide no guidance whatsoever:
Xanathar's Guide to Everything (p. 77) provides further guidance on falling, but nothing that would allow a pegasus to catch you:
Using these optional rules, you instantly fall up to 500 feet, which does not allow for your pegasus to swoop down and catch you.
I have used a house rule for this.
I played a bard that used find greater steed, and the first time I got knocked off, my DM and I agreed that there should be some chance for my pegasus to do this. Here is the rule we used:
The idea here is that both the rider's skill (Dex save) and the mount's agility (Dex check) are taken into account, and it costs you some altitude to remount. For falls under 30 feet, there isn't enough altitude for your mount to catch you. For us, this didn't happen often enough for it to slow down the game, and we settled on (30 minus Dex save) as a good DC; you can adjust that 30 to be higher or lower if you want it to be more or less difficult.