I don't see any reason that you couldn't, but it will be very difficult (DC 20 Acrobatics just to get 5 feet up with a running start). As far as the grappled creature falling or flying away, you'd probably need to look at its encumbrance limit:
Lifting and Dragging: A character can lift as much as his maximum load over his head. A character's maximum load is the highest amount of weight listed for a character's Strength in the heavy load column of Table: Carrying Capacity.
A character can lift as much as double his maximum load off the ground, but he or she can only stagger around with it. While overloaded in this way, the character loses any Dexterity bonus to AC and can move only 5 feet per round (as a full-round action).
It is theoritically possible to jump and grapple a flying creatuer, but only a monk really has a chance at doing so.
The height of the monk is irrelevant, only their size. A medium creature fits in a 5ft cube, regardless of wehther they're a 6ft tall human or a 4 ft tall dwarf.
Based on your description the monk needs to jump 10ft up. A 10ft jump is DC 40.
And once you get up to him, you now have to grapple. Which is all well and good, but the flying creature doesn't necessarily fall. He will take the penalties of grapple, but so does the monk.
Grappled: A grappled creature is restrained by a creature, trap, or effect. Grappled creatures cannot move and take a –4 penalty to Dexterity. A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple. In addition, grappled creatures can take no action that requires two hands to perform. A grappled character who attempts to cast a spell or use a spell-like ability must make a concentration check (DC 10 + grappler's CMB + spell level), or lose the spell. Grappled creatures cannot make attacks of opportunity.
Now, if the creature can support both their weight then he continues to fly there (assuming he can make the fly check to hover), unable to move until he breaks the grapple.
I have never heard of this 5' cube. A 6' character cannot stand in a place with a 5' ceiling.
A good way to rule it for trying to grapple would be to use the height of jumper. That would allow for extra reach to really grab hold.
So in this case, the DC of 40 would be correct.
A limiting factor for jump + grapple would be to ensure that the jump distance is not more than the creature's movement rate, as grapple is a standard action.
Any aoo will caused by jumper for acrobatic use?
Only if he passes through the target creature's threatened area.
If the target does not have reach, the attacker can enter the threatened area and initiate a grapple. No AoO.
If the target does have reach, the attacker will have to pass through the 10 foot threatened square to get up to 5 feet to initiate the grapple. That provokes an AoO.
Missing the creature (as in failing your grapple check and not colliding) and falling like a sack of bricks will also most likely invoke an AoO (mostly because your fall will pass through more of the opponent's threatened squares).
Missing the creature (as in failing your grapple check and not colliding) and falling like a sack of bricks will also most likely invoke an AoO (mostly because your fall will pass through more of the opponent's threatened squares).
You're right: if you fall away from the target, you'll leave a threatened square and provoke.
In one game last week, we had a flying bad guy, and our(Hunter riding a giant wasp) was flying behind and above him. Our high mobility, leaping Rogue jumped up into flanking position and attacked, then provoked and AoO as he fell. The summoner though this was awesome, so his eidolon jumped up into the same flanking square, attacked, and fell to the ground (he provoked an AoO, but the bad guy didn't have combat reflexes). Then the Hunter attacked, and knocked the bad guy to the ground, right on top of the eidolon...
We were all highly amused.