The gun control issue is one of the more complicated issues. With something like SSM or even tax issues you can pretty quickly come up with an opinion even if you are not well educated on the matter. Gun issues are a bit more complicated. Gun experts and LEO in general seem to be more gun-rights so that's where I lean. My POV is that rather than restricting what types of guns people can have we should restrict what kinds of people can have guns.
People who DEFINITELY should have limited or no access to firearms:
Good news for you, as I'll address further, most of what you want is already law to and extent. Though that can be refined a bit. I'll explain.
1. Convicted violent felons, or at the very least felons convicted of an aggravated violent felony
Anyone with a felony conviction or dishonorable discharge is already disbarred legally from owning a weapon. To refine that I would rather that the felonies be only those that are violent or have violence as a component(black market felonies) and I would remove dishonorable discharge, they are both catchalls that overgeneralize the removal of rights. If they were shortened in scope to eliminate non-violent offenses and had a period certain opportunity for the restitution of rights for good behavior it would be perfect law.
2. The mentally insane. Of course it depends on the disorder, someone with a Cluster B personality disorder (this includes psychopathy, which is the primary reason I chose Cluster B: psychopaths should have NO access to firearms as they are incapable of responsible or non-criminal use of weapons); it also includes borderline personality disorder (excessively emotional), histrionic personality disorder (excessive pathological attention-seeking), and narcissistic (arrogance with a touch of total lack of empathy)). Paranoid schizophrenics should have LIMITED access to guns because although unlike Cluster B they are not inclined towards violence, they cannot differentiate between reality and imagination and may hallucinate a burglar and end up shooting their neighbor. While registering to get a gun (something I believe could be necessary)* people should be given a psychological exam to see if they are mentally fit to own a gun.
Anyone involuntarily committed to an institution of mental health is disqualified from owning a gun, I like your idea of only limiting mental disorders to those of violence or potential psychotic breaks, makes more sense than someone within the fully functioning autism spectrum being disbarred "just because". I think upon conditions of medicating the problem within control some people could have their rights restored. But I am not willing to go as far as mandatory testing for everyone who wants a gun, it's a little too far on the prior restraint side for my tastes. Rights do have some responsibilities and risks.
People who reasonably could have access to firearms restricted:
1. People with ties to organized crime groups like Mexican drug cartels or Mafia
2. Non-citizens. Illegals should be given no access to weaponry.
No argument to your first group.........but you have to prove it, street gangs would be the most preferable group, they are the most random in application of violence. You have to go out of your way to be on a Mafia hit list, and cartels are a combination of being in the wrong place at the wrong time OR being in their way. Street gangs will shoot at the wrong address and call it a write off.
If I were to restrict weapons I would say off-the-bat no WMDs (nukes, bio weapons, chem weapons), no explosives. If I had to restrict any firearm in particular it would be a sniper rifle. Reason being is that's past the reasonable point for self-defense - a 14.5×114mm cartrige could be fired over a mile away. I don't think people should be given access to guns like that in the highly improbable case of a revolution against the government. At that point the law is void anyway.
Okay, no problem with restricting WMDs, they are indifferent weapons, area effective, and you don't aim them as much as disperse them. Explosives have a purpose, and a semi-safe handling, they shouldn't be outright legal but permittable, I have no problem with an advance license requirement for ownership and detonation. "Sniper" rifles are really just long guns, there is nothing about them that makes them deadlier than a hunting rifle, most large bore hunting rifles are accurate from +705yds to a little over a mile, with the longest shot ever placed on target at about 2mi. by a Canadian sniper using a Barrett .50cal. but just about any long barreled hunting rifle will shoot up to about a mile and on target if you know how to adjust for windage and loss of velocity.
*I understand that criminals could find ways to get arms without registering, which makes me a bit iffy on registration. I think registration should be simple: a psychology test, background check, etc. It should be ABSOLUTELY FREE to register although a minor tax on automatics might be a little useful. Since registering would be free it would be a FELONY to get unregistered weapons in the United States of MadLib.
Criminal background check, IF there is a further question by civilians(such as the gun range owner in Colorado) who notice behaviors that are a little off allow for them to sign a sworn affadavit, legally binding, that allows for further questioning/testing.