It will play quite a few A500/A1000 games but the newer processor (68020) and the AGA chipset will cause some incompatibilities, especially those games that do a lot of hardware bashing.
Look into WHDLOAD. It's a software compatibility layer that will allow you to play 99% of A500/A1000 software on your A1200. The price is reasonable and it's still supported by the author.
I'm not familiar with any C64 emulators out there for the A1200 but they do exist and from what I hear they do an impressive job at emulating the C64/128.