If I buy a ps3 game online and then buy a new ps3 can I transfer the games from the old ps3 to the new one?

I'm renting a ps3 and I plan to buy a new one next week. Will I be able to put the games I've bought online onto the new ps3 or will I have to buy them again?

How can I download and transfer free games from my PC onto my Helio Mysto?

I found a website where there are many free games for the Samsung phone, which is the Mysto phone itself. When I download the file and transfer it to my phone, the game is not supported by the phone. Any ideas on how I can get a few games there?