Can PSP Batteries be recharged via USB?

I don't think the first gen can do it but all others should be able to. There is a setting in the Settings menu for USB Charge, so first you need to make sure that's on. Not sure on the effects on battery life, but it should be the same as any other forms of charging.

I have a PSP 2000, and I have a USB charger for it and it works just fine, has for years. I do not know about the first gen but I don't see a reason why it wouldn't. There may be affects I do not see or notice, but my battery life is as long and full as it ever was.

From memory you are able to charge it when you have a firmware update. I don't remember which firmware allowed that functionality. You never know unless you try it.