My 360 is modded and I have no issues with it. When you get banned on live they ban the serial number of your console and not your actual live account., at least this is what was happening the last time I was using live, which was awhile ago. I didn't get banned I just moved and internet was too expensive for the 1st few months so I just got it back because highjacking the neighbors wireless was too slow.
With the burnt games you need to have an ISO program to burn them and you should use a bit torrent to download teh games but if you have an ISO program you can just rent games and then burn them, when you burn them though you need to use dual layered DVD's and I would suggest an external HD so you can save all the games you have downloaded or copied to that.
I get most of my information from the xboxscene forums if you want info I would go there.