This is completely fine. Most (more than 95% I'd say) handheld devices today are made to do that. The charging systems in all of these are designed to not only stop charging when the battery is full, but charge more than the device usually uses up. In some cases the battery may be used faster than it is being charged if you are doing huge tasks on it, but this is very rare.
The worst that will happen from doing this is the charging process will be slowed because some power is being used while it's being put in.
Well if it has no battery, then nothing is actually 'charging', but, yes, the PSP can be played while the charger is plugged in, even if there is no battery.
yes you can just play as normal
Not at all.
yes cause you can play ds and dsi while charging
No you can not play the PS Vita games on a PSP.
The Vita is the new Playstation portable that will not be released until December 31 2011 You need the Vita to actually play games on it
Yes, the PS Vita's games are region free. A Japanese PS Vita will play an EU game.
no
No, but you can link the Vita to the PS3 console
You should buy a PS Vita if you want to play PS Vita games.
There is an increasing selection of PS games, 'PSOne Classics' available on the PSN, the PS Vita has an emulation mode to play them on. Also, there are some that aren't available on the PS Vita's PSN store but are on the PS3's, which can be transferred across to the Vita via the PS3.
The iPod/iPhone/iPad charge cables, in both its legacy and lightning variety, are not compatible with the PlayStation Vita for charging or data transfer purposes.
No, the Vita's browser can't play flash-based games.
None the Vita can not play a PS3 game
The vita can be a controller for the PS3 like the PSP could, but does not have the power to actually play any of the games.