It can play any PSP game that is available on the Vita's PSN store. More games are being made available though it does not have access to the full PSP library just yet.
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The PSP 2, or NGP, will not have a UMD drive, so any games you have in UMD format will not be playable on the PSP 2. However, any games that you have downloaded on your PSP 3000 can be transferred to the new PSP.
The PSVita does not support the PSP UMD games, but does support a very large percentage of digital PSP game purchases.
No, it does not. It uses its own game format, the games are on PSVita game cards, they're like memory sticks.
Yes, but not with the PSP's Universal Media Disk (UMD). You can download them off of the Playstation Store, though. The PS Vita will have a different form of media that Playstation says is better and can store more memory.
The Playstation TV is pretty much a screenless PSVita. It can play a selection of compatible PSVita games and has a slot to insert them, as well as certain PSVita games on the PSN.