First off, you must have a PS3 on FirmWare 3.55, or lower, to mod it. If you are on a lower FW, then simply upgrade to 3.55. However, if you're on a higher FW, then you must downgrade using a hardware Flasher. Below is table for what models are downgradable and what flashers can be used on those models:
|PS3 Model #||Downgradable||Progskeet||E3 Flasher||Teensy++|
CECH-A to CECH-G
|CECH-H to CECH-Q||Yes||Yes||Yes||Yes|
Basically, all fat models are downgradable, if you have a NAND model (CECH-A to CECH-G) use Teensy++ and if you have a NOR model (CECH-H to CECH-Q or any downgradable slim PS3) use E3 Flasher. For downgrading I suggest you follow a tutorial: Using Teensy++/Progskeet or Using E3 Flasher
You can connect to PSN while on CFW, but there is a high risk that your PSN ID and Console ID will be banned. There is a way to unban your console though... it requires someone else's Console ID and if that gets banned too, you'll end up banning them.
And as for emulators, RetroArch has a PS3 port, so I'd try that out.
Also you can emulate PS2 games on PS3, any model PS3, however the emulator was never fully completed (it's Sony's own, they gave up on a full PS2 emulator for PS3 and instead now repacking PS2 ISOs with a modified version of the emu for $10 on PSN) so not all games work. Check the this list for games that do work.
PSP games also work on PS3 through the same method as above, though not many games work (less than that of PPSSPP).