Yes you can, the PSP battery will continue to charge even when you are playing.
I would not recommend to do that, it is better if you leave it to charge and then play.
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play the ps3 games in psp they will converted to psp games
the game has to be game share compatible. you basically got to gameshare on your psp and then your friend loads the game up n his psp.
not imported ones
Nope, but it might be in a while. Probably not though, because its a online game and PSP u can play with friends, but its not really a online console
If you have residual charge on the battery to power it on and updated to the latest version of firmware for your PSP, certain PSP models are able to charge over a USB connection - but it may stop charging if you try playing a game on it while doing so.