#2. In order to impress the locals, Litefoot wants to do a backward somersault off the bar table and throw his dagger to hit the bull's eye on the dart board 20 feet away...all at the same time.
There are only two methods of attacking while actually moving (in cinematic terms). One is with the spring attack like form - using spring attack, shot on the run, ride by attack, or flyby attack. The other is mounted archery. Every single form says that the person resolves their attack some point during their move but after a certain amount of movement and before the next portion of their movement. In other words, the person is considered to be in a certain position while attacking.
For example, mounted archery dictates that all shots fired are done at exactly half the double (or run) move of the mount. Spring attack says the attack allows a move, attack, move. Clearly, one must be stationary at the time of the attack during the attack (that is not to say that the person is stationary during their turn.
edit - there is atually a third form of movement - moving after casting a spell that holds a charge as is the case for a touch attack. Just added for thoroughness.
Therefore, for someone to pull of this amazing feat (no pun intended), Litefoot requires shot on the run. What he is doing is tumbling both before and after his attack (during as well, but ruleswise he is considered stationary). So he has to roll a tumble check for the portion of his move before his attack and tumble check for the portion of his move after his attack.
The tumble check ought not to be obscenely difficult. Afterall, anyone who knows how to tumble ought to be able to do a backwards somersalt. Although, I would imagine a back hand spring would be more along the idea of the poster, as a backwards somersalt would mean he was rolling on his back off the bar top. Of course, a bartop is not more than 5' off the ground and this would prevent the additional tumble check to land after falling 10' (which would happen if the Litefoot did a handspring). But regardless, standard tumble check probably at DC15. I would introduce some circumstance penalties depending on the environment. If there were alot of people in close quarters and he was effectively tumbling through another's space the DC would increase to 25 (as is standard per the PHB). Of course this is going to be alot easier than tumbling through the space of someone actually trying to attack him so I would probably introduce a circumstances bonus in either case of somewhere between +2 and +4 depending on how many people were around and a circumstances penalty 0f -2 to -4 for the amount of clutter (barstools, ale pints, beernuts, etc.) So after the initial tumble check to get into the air he then has to make an attack roll. This would be the attack portion of shot on the run.
The attack roll would be vs. a miniscule stationary target (sorry IDHMBWM). I don't know the exact AC but I think it is something like 10 (base) -5 (stationary, effectively 0 dex), +8 or+9 (for the size). Also, there is the range increment penalty (since the dagger has 10' range incment so that would be a -2 on the attack). I would also introduce another circumstances penalty for the difficulty of the maneuvar. First off he is throwing a dagger at a dart board Depending on the dart board I may also introduce a -1 cirucmstances penalty because some dart boards have metal round each section ( and for the bull's eye, that dagger is going to have to be at the dead center of the bull's eye or be angle there because if it's not the metal will deflect it - this would not be the case for the dart because dart's are so much smaller than a dagger). Also, given the akward angle of the attack, I would probably apply a -2 to -5 to the attack roll depending on what orientation the character wanted to be in while throwing the dagger. If he wanted to do it from behind (while not looking at the target, the target would effectively have 100% concealment giving a 50% miss chance). Then the character would have to roll another tumble check to land ok. This would be again at DC 15 unless he went for the handspring and was trying to effectively land safely after falling 10'.
So
Tumble check one DC 15 or 25 base with modifiers for circumstances (-2 to -4 and +2 to +4).
Ranged attack roll at at least -2 (+ a -2 to -5 for circumstances) at AC of 13 or 14 with a possible 50% chance miss rate.
Tumble check two DC 15 or higher base depending on height with similar modifiers to check 1.
This would be simple enough for a mid to high level character which it should be, pretty difficult but not impossible for low level characters which it should be. I would hope that a low level character who focused on throwing weapons would be able to pull of this maneuver because it is not all that difficult.
If successful I would give a circumstances bonus to all chr based social interaction skills of anywhere from +2 to +8 depending on where Litefoot was. If he was in waterdeep, probably only a +2. However, if he was in a small farming village where the most action the farmers see is a bit of rain, I would probably go for +8. This would be most useful for diplomacy or intimidation and I would probably also houserule that no matter the checks on either of the rolls, the characters starting status would be at least one more category more favorable than the NPC started out at. (Just because the roleplaying was so cool and I try and encourage cinematics). Of course that would be houseruling.
Alternatively to the throwing the dagger while moving I would also offer the possiblity to the PC of attacking moving and attacking. Throw a dagger before the somersalt which would be during the surprise round and then roll initiative (most like the PC is going to win vs all the farmers at least) then do the tumble and throw anoher. This or some version of it would alleviate the problem of not having shot on the run - which many characters do not have. The tumble check would the be same as in part one. The last option would be to do this without shot on the run. However, doing so would be difficult for the PC because he would make the first tumble check then the attack but he would then be considering falling after the attack and even if he made the tumble check, it would be described a lot different than a character with shot on the run would be described and I would probably halve all of the circumstances bonuses received from success.