I think he was asking for specific values for the DC checks.
Leaping over an opponent of the same size to strike an opponent behind.
I would make this a Tumble DC 25 plus the apropriate jump DC for that height. Maybe grant a +2 circumstance bonus to the attacks due to surprise factor the first time in combat.
Tumble DC 25 lets you through the opponent's square w/o AoO. The "jump over" description is just window dressing; if he's got the ranks to be able to do it, let him describe it that way. Too many rolls slow the game down.
If he can't/won't tumble, use a standard jump check, and take the AoO if he succeeds. A failed jump check is probably resolved as a (charging) bull rush attempt against the jumpee (again with the appropriate AoO).
Me-doken! 
Springing off a wall to make an attack.
Tumble DC 15 +5 for every 5ft above 10ft.
I'm not sure this requires a check, other than what would be required to move this way (an appropriate Jump check, if trying to reach a specific height to avoid attacking through partial cover provided by a low wall, for example. Or if the off-the-wall movement is being used to avoid another foe or obstacle.)
Otherwise, again, it's just a descriptive move/attack. I wouldn't penalize the player by making him succeed at a tough roll just for the sake of livening up the game like this. I'll take well-described actions any day over "I move up 15 feet and attack. Does AC 19 hit?"
...all this "no roll" stuff is assuming the character is good at this. If it's the dwarf tank with a -6 total Tumble modifier, then it might be different. But if you can make DC 5 without rolling, go for it.
Jumping off of an opponent to reach a higher location.
Obviously this causes an AoO. Add +5 to the Jump DC.
I'd call it an Overrun attack, but make a Jump check (with the appropriate adder for the foe's height) if the melee touch attack succeeds, rather than resolving the trip attack as normal. Since you aren't trying to charge through his space, I wouldn't let the opponent choose to avoid you, as would be normal for an overrun. (Spring Attack or a DC 25 Tumble check is needed if you want to avoid AoO due to moving thru threatened spaces like this.) A penalty to the jump distance could be appropriate, but making the touch attack and sucking up the AoO is probably penalty enough. (Without some magical enhancement, vertical jump checks suck cream-style corn anyway.)
If you aren't charging in, I'd say the Jump check could be made by first successfully grappling the foe. Pulling his helmet down over his eyes first is optional.
Fighting an opponent while hanging by your legs on a crossbar.
Wow, this I don't know. Maybe a Climb check. Obviously a hefty attack penalty unless you have the Brachiation feat.
Fighting an opponent while hanging by your legs on a rope.
Again, don't know.
The Brachiation feat is related to movement speed, and requires free hands, so I'm not sure it would apply here.
The Climb skill covers the defensive aspects. DC 10 (beam/ledge) or DC 15 (unknotted rope) to hang on if you take damage. You don't get shield or dex, and opponents get +2 to hit you. Since you aren't holding on with all limbs, an additional -2 circumstance penalty would be appropriate.
Offensively, there's nothing I could find in the books that makes it hard to attack from these situations. Attacking from prone is -4, but kneeling carries no penalty. Attacking while levitating is -1 (up to -5 for continued attacks). These two climbing situations aren't quite as limiting as lying flat on the ground, but they're more limiting than the complete freedom of motion, but lack of leverage, gained while levitating. So, for a cinematic game, no penalty. For a serious game, go with the standard -2 circumstance penalty to attack rolls. (There's an argument for ditching the Str benefit while hanging on a rope due to lack of leverage, but if you don't lose it when levitating, you shouldn't lose it when climbing.)
Fighting an opponent while on a beam.
This is handled in the balance skill. You need 5 ranks at least not to be caught off balance (+2 attacks against you and you loose dex modifier to AC). Every time you´re hit you make the apropriate balance check not to fall off.
Agree.
