MadCatz released, along with arcade sticks, a FightPad. I haven't used it myself, but I've read good reviews.
No sorry, but Sony have said they are aiming to make as many compatible as possible. Heres a list of all the initially compatible games.
I found this FAQ which answers a few questions but I'm not sure if this is the definate answer you were looking for. It says (about half way down):
Q: Is PS Vita compatible with other PlayStation platform games like PS3, PS2, or PSP? Do you have any plan to support other PlayStation platforms in the future?
A: Users can play PSP titles, minis and PSone classics titles which are offered on PS Store. We do no have any plans to make PS Vita compatible with PS3 or PS2 titles at this point.
Q: Is PS Vita compatible with PSP content other than games such as video and comics available on PlayStation Store?
A: SCE plans to make those content available on PS Vita.
Q: Can PS Vita users continue using their save data for PSP games that they have also downloaded from PlayStation Store to PS Vita?
A: We are looking into the opportunity to do so.
The bolded would suggest no, only confirmed titles would be supported however this is really vague and still doesn't completely prove or disprove that the use unsupported games will work. However Sony have already said UMDs aren't supported and if the only way to get the PSP games onto your Vita is through the store then I highly doubt any unofficially supported games will work.
Sorry for not providing any concrete definiative sources I just couldn't find any.
Best Answer
No, not all games are compatible. Wikipedia says that the game you mention is mostly playable and can be completed albeit with some difficulty: http://en.wikipedia.org/wiki/List_of_Xbox_games_compatible_with_Xbox_360