1. To hack your PSP, you need a Pandora Battery and a Magic Memory Stick. Look in the Guides section for tutorials on how to make both. I recommend Catb50's Magic Memory Stick maker as it is extremely easy to use, and works great.
2. To put ROMs on your PSP, you need to download an emulator for the ROM's system. Once you do, drag the folders containing the emulator into the GAME folder in the PSP folder on your memory stick. If that doesn't work, try the GAME150 folder.
3.Utilities and homebrew are the same, download the files, drag them to the GAME folder in the PSP folder of your memory card(X:/PSP/GAME) where X is the drive letter of your memory stick.
4.By connect to the Playstation Network, do you mean, play games wirelessly, or do you mean access the PSN store? You can definitely still play games wirelessly. I'm not so sure about accessing the PSN Store, I think you can, but I'm not sure.
5. To play the backups, simply go to the Game section of the XMB, go down to the Memory Stick icon(last one) and press X, then just select the game.
6. I'm not sure about connecting to the PS3 as I do not have a PS3.
7.There are plenty of Guides out there that will help you with all of these. Go to the Guides section of this site, and find the one that's right for you. There are ones relating to hardware mods, and software mods, among other things. Everything is there, as long as you know where to look