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You can with a USB cable
Some firmwares allow the PSP to charge from a USB connection to a PC.
A PS3 will charge any device that can charge from a USB port. This includes, but is not limited to: PSP, Ipod, and many cell phones.
Yes, the chargers for each model are the same. It can charge a PSP 2000 or a 3000, etc.
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.