RGB is the pure color signals on 3 different cables(red, green, and blue). Most TV's and such in the US don't have RGB inputs.
I would start by cleaning the cartgidge contacts and the cartridge slot of the Genesis. The part about having to reinsert them over and over makes it sound like a good idea anyways. It is also possible that the batteries have gone bad. They are going on 15-20 years now. You can of course replace the batteries in them. It's a bit of work if you're not comfortable with electronics but it's not too hard to do. You can search this forum for plenty of info on how to do it.
The poor video signal sounds like an RF issue to me. What happens if you wiggle it, move the wires around, etc...?