The relevant text is this:
While you are in this stance, you gain a +10-foot enhancement bonus to your speed. If you move at least 10 feet during your turn, you gain a +2 dodge bonus to AC until the beginning of your next turn.
So let's go through your questions:
You get the AC as soon as you move 10 feet
Let's say you wanted to move 20 feet, but moved 10 feet and provoked an attack of opportunity. Did you move 10 feet this round? Yes, so you get the AC bonus. Is your move action complete? No, you get to keep moving after the AoO is resolved.
You get the AC if you change into the stance after moving
The stance does not say that you must move 10 feet while in the stance. The requirement is only this: did you move 10 feet this round or not.
You get +4 to jump most of the time
The Jump skill reads:
If your speed is greater than 30 feet, you gain a +4 bonus for every 10 feet beyond 30 feet.
It does not discriminate between sources of bonuses. So if your speed is 30 and you activate this stance, you get a +4 bonus to Jump. If your speed is 25, and you activate this stance, you gain no bonus, because your speed is not 10 higher than 30. If your speed is 20 or lower, you reduce your penalty by 6 instead.
If you have another enhancement bonus to speed (say, from levels in Monk) then you don't actually get a speed increase, because the bonuses do not stack, and thus you get no Jump bonus.