Can you replace a Peg Perego Four Wheeler 12 Volt Battery with a different brand? (Like a 12 Volt 20Ah Rechargeable Battery....will that give it more power, or last longer??)

Yes!!!! you should use SEALED lead acid batteries, not golf cart or boat batteries due to risk of build up of gases inside the little chambers and possible explosion or acid burns. batteries from computer back up units are great and you can find pretty good deals on ebay for those. keep the connectors off the old battery and use it on the new one. the one possible downside is the bigger battery might not fit in the regular spot, but you can modify the space usually with a few strategic cutouts. good luck!