Run a PSP 3004 without battery

From this Gamespot article

The PSP-3000 plays the same games and offers the same media support as the older PSP models

So all the games are backwards compatible.

From Wikipedia

the PSP-3000 ... has an improved LCD screen with an increased color range, five times the contrast ratio, half the pixel response time to reduce ghosting and blurring effects, a new sub-pixel structure, a microphone, a new disc tray design, new button designs and logos, and anti-reflective technology to improve outdoor playability. It can also output all games by component or composite using the video out cable.[77]

So it's better. The biggest improvements seem to be the reduced ghosting (which still exists but is well, reduced) and the anti-reflective technology.

1) Not to my knowledge as it currently stands unless the game ALSO states its playable on the PS3

2) I assume you want to hook the PSP up to the PS3 to play the PSP game on the PS3 with the PS3 controller, This is also a No (You can remote in to a PS3 and play games on the PS3 that stream to your PSP... But I so far only have done this with games I have purchased digitally like Wild Arms 3)

3) The Slim PSPs (rev 2000, 3000) have a cable you can buy that allows you to play the games on a TV (Go search Amazon.com and Id suggest the Sony cables).