Quoting DumNDuMMer

I was wondering if anybody could help me with a few PSP/PS3 questions.

1.) Can I use some type of homebrew to play PS3 games or do I need a mod chip? Like backed up copies from originals? Is there even a mod chip available? Kinda like how you can dump a PSP game iso and then put it in a memory stick and then play it on the PSP without the UMD.

To play homebrew games, you'll need either a PS3 dev kit (expensive) or a mod chip which will certainly void your warranty. But it's your console - as long as you don't do anything illegal (pirate games) you should be fine. I should also warn you that a firmware update could potentially brick modded systems. I haven't personally seen it happen, but it is always a possibility.

Quoting DumNDuMMer

2.) Can I play PSP games on a PS3 from a memory card or any other way or am I better off finding a cable/plug of some sort to connect the PSP directly to the TV?

I highly doubt that's possible. Remote play works PS3 -> PSP, but not the other way around. You'll need to try a cable to hook directly into your TV.