Mental health checkups would include a psychiatric evaluation.
A mental health checkup IS a psychiatric evaluation.
Beating up your spouse or having a violent temper can disqualify you.
Beating up a spouse is a criminal offense and disqualifies a person from owning firearms.
Being drunk can disqualify you.
Forever? Just while intoxicated? Are you proposing that people be constantly monitored to determine if they are ever drunk?
If they failed, they could still buy a car and failing the test would disqualify someone from buying other weapons.
A car is certainly a very effective weapon as has been demonstrated more than once. Why would you allow a person who isn't qualified to own a firearm to own such a potentially dangerous item?
In a nutshell, if a person passed a universal Background check, passed the psychiatric evaluation and passed a written test and the police are satisfied that the person passed these requirements, the police cannot deny the person, the conceal carry permit, which is shall issue.
Why do you believe that, unlike other constitutional rights, exercising this one requires mental health check ups and background checks. Do you support such measures for people wishing to exercise their right to free speech or their right to vote?
If you believe that guns are in a different category, explain why they are in a different category and, how from that it follows that people who legally own guns should be subject to such extreme intrusion as being forced to be mentally evaluated periodically.
What sort of mental health problems would disqualify a person from gun ownership, how would those conditions be assessed and quantified to determine if a disqualifying threshold is reached? Who gets to set those thresholds? If a person is too unstable to own a firearm, why are they safe to remain in society where they can operate vehicles, access or make non-firearm weapons (kitchen knives, clubs, homemade explosives, firebombs, etc.) to use and carry, interact with the general public? If a person is stable enough to drive on the public roads, follow the laws, vote and interact with the general public, why wouldn't they be able to own firearms?
What other rights should the government be able to restrict without any criminal proceedings?
You want to know why people are against some of the things you propose, but when questioned, you simply repeat your position again without actually explaining how it makes sense. Explain how it could be implemented in a free society without horrific invasions into person's lives. Explain how one could justify committing such acts against people who haven't committed any crimes. Explain how it makes sense to allow people who are allegedly so dangerous/unstable that they can't own firearms to otherwise run free in society where they can easily create all kinds of havoc.