is it ok, if your at home or something, to play with the power adapter plugged in so you dont waste battery? if doing this, should the battery be in the psp or be taken out to avoid unneccesary charging since the battery isnt depleted?
Most things won't even turn on without a battery, and the ones that do get reeeal hot after a few minutes. I just wait until my PSP battery gets low and then charge it, whether I'm playing at the moment or not. I unplug it as soon as I see that it's charged. It's probably not a good idea to be charging it all the time, that supposedly shortens the lifespan of rechargeable batteries. Basically, just charge it when it's low or when you're going to be taking it somewhere, and don't leave it plugged in.
after fully charged, you shouldn't leave your battery in when using ac adapter. it will shorten your battery life. after the battery is charged, it goes into so-called "trickle mode", where small amount of battery charge is used and then recharged back to full again. it is very damaging to the battery. this is why all electronics have auto-shutoff after it's been fully charged. if you want to use ac adaptor, make sure you take the battery out.