hello fellow interested-in-making-visuals-with-nes buddy!
First of all I second your questions about flash cartridges because I still don't know about that.
Regarding the other questions,
I believe that if you have a NES with a composite video output you can hook it up to a cheap HDMI or VGA converter (~10$ on chinese websites).
You can indeed modify (glitch bend) a NES! I made some ugly modifications on my first attempt but they are quite curious and interesting, you can check them out on my website, otherwise check out casper electronics' mod (his website is currently down tho) or bitcrusher's mod. I also made an arduino based glitch step sequencer.
About mixing visuals with something else, my current option is using an old hardware video mixer (panasonic ave5, pretty popular among vjs) which I'll probably also try to bend. Hardware video mixers are pretty rare and expensive though, so you may want to try other options like recording a footage with a DLSR or an old camera (if you have one) of your screen and then prepare clips via software and use only a laptop.
You can consider a "Karl Klomp" dirty video mixer but it only works on analog tvs.