A massive registry has several problems. From a practical perspective, it requires a lot of time and energy. If you have a registry, then there has to be a penalty for people who don't register or its pointless. If you penalize people for not registering guns, then it winds up intruding on a LOT of people's privacy for no good end.
And, is a registry really going to be useful in preventing violance? The basic answer is that it will provide a way for the government to come in an seize guns from legal gun owners, basically law-abiding citizens -- like they did in New Orleans and like they have done historically in the past in other countries. A registry does NOT do a thing to prevent crime, not really.
Back ground check is reasonable. I would like to see a few of the existing loopholes sealed. Buying a gun at a gun show, at a minimum, should require a check. Tracking private purchases would be more difficult, but is something worth considering if we want checks to work. Similarly, I can definitely see cause to add some people with mental illness, BUT we have to be very, very cautious here. People tend to target folks who are just "different" and see them as being some kind of threat.. even if the real evidence is that 99.999% of those people are not a threat. To be effective, a registry of people with mental illnesses needs to happen only after a lot more research. The current approach of targeting behavior is probably better overall. We are not yet able to really predict who will or will not wind up being a killer... with a few exceptions.
Exceptions to the "we don't know who will be a killer". I would like to see limits placed on people who abuse animals. People who actively abuse animals (not just neglect them or "hoard" them, I mean people who torture them for fun) often wind up hurting humans next. IN some cases, I think juvenile convictions in this regard.. and certain kinds of rape ought to be carried over into adulthood, though with a great deal of caution and some stop gap measures to ensure that a kid isn't, say targeted for life because he got mad at age 10 and hit a dog. On the other hand, if he had a habit of collecting neighborhood strays and doing "experiments" on them... well, maybe. Similarly, we already target the teenager who basically pissed off his girlfriend's parents a bit too much, but the boy who rapes his 5 year old neighbor at age 11 might need to be tracked.. or at least, the file should be tagged so that it might be visible IF other events happen later (if he is convicted of other rapes, etc.). Anyway, those types of people probably ought not to have guns.
People who knowingly try to buy a bun when they are on the "do not buy" list ought to be at least tracked. I am not necessarily comfortable with putting someone in jail for trying once.. particularly if there is any chance that there could have been some confusion on the part of the person applying as to whether they were eligible or not. However, if someone tries again to buy a gun after being turned down.... that ought to engender some penalty, perhaps jail.
Other than that, I think the concentration should be on better mental health care, and better training of youth about positive gun use. When the only association people have with guns is violance, then of course they are used for violence.
Along those lines, gun owners have to take more responsibility for teaching and using guns safely. The stray bullet that hits a pregnant woman in her car in her driveway hurts every hunter -- no matter how reasonable the use was deemed by the law.