So how does ATC check aircraft registration?
They don't, that isn't their job. The only time that anyone will ask to see your registration in normal circumstances is during a ramp check, and those are done by FAA inspectors. There are probably some other 'routine' checks, like a checkride examiner verifying it as part of the checkride or your insurance company asking for a copy. I suppose it's also possible that your state might want to see it for some tax-related reasons.
Do they have live access to the registry?
I have no idea, but as Greg pointed out the registry is open to anyone with an internet connection so it's at least possible that they do. But I don't really see how any of the registration information would be relevant or useful to ATC operations anyway. One possible exception could be search and rescue: I guess the registration information might be useful for contacting the aircraft owner if the aircraft is missing.
When someone calls an ID over the radio are there controllers that routinely check the ID's registration?
Again, I have no idea, but I can't think why ATC would want or need to check it and I couldn't find anything in the ATC orders about checking it.
Is it illegal to make up an ID for radio interaction with ATC?
I couldn't find any specific regulation about that, but if the FAA wants to enforce it then I'm sure they'll find something. They could always fall back on 91.13 and claim that by lying to ATC you're displaying reckless behavior. And if the government really wants to get serious with you then they could charge you under 18 USC 1001: lying to a federal government agent is a criminal offense.
Are there people that visually watch for tail numbers and discrepancies with radio calls?
Yes, they're called planespotters :-) I'm only half-joking: some people are obsessive about things like that, and it wouldn't surprise me at all if someone out there likes to look up the owners of aircraft he hears on the radio.