I just bought a PS2 about 2 weeks ago and haven't researched it yet. You have to get a mod chip in order to play burned games. From what I have read, it appears that you will still need to do a swap trick with the DVD's. I believe burned PS1 games will work on a modded ps2 without swapping. I have a modded ps1 so it doesn't matter to me. I'm going to look into it further this week and get the low down. I'll post the information for you when I find out.
I've seen external USB mod chips for the PS2. These are real nifty if you don't feel comfortable taking apart your PS2 and soldering a few wires.
I've never ripped a DVD, so it's probably a lot more complicated then going to blockbuster, renting them, and copying them. You'll have to format the dvd games in order for them to be burned onto a normal cd-r. A lot of companies started using copywrite protection on the ps1 games that made things more difficult and I'm sure they took more precautions with the PS2 discs. Luckily I have a friend who runs a T1 dumpsite and it's loaded with PS1, PS2, DC, and PC games. They're ripped and ready to burn, which makes things a lot easier on me.