Phat PSP and Slim PSP batteries can be made into Pandora Batteries. If you were to buy one 95% chance it'll be a phat PSP which works in Slim PSP fine.
eBay is a terrible, terrible place to buy anything PSP related. If you need a Pandora Battery, buy a Datel TOOL battery rather than get scammed on eBay. Aslo never buy Memory Stick from eBay, 99% of them are fakes.
Pandora isn't needed when you upgrade to next firmware, its only used to get you onto custom firmware. There is also no need to switch between official Sony and CFW.
CFW is the official Sony FW, acts and looks the same, just with some extra additions.