I've proposed a lot of things in numerous threads. Too numerous to repeat here.
Actually you were kind of sparse on a lot of stuff. You listed problems: FFLs who break the law by selling to prohibited people, stolen guns on the street, mentally unstable people owning firearms, universal background checks, ...
Specifically what do you propose?
a) FFLs who break the law. What do you propose there, make it illegal? It already is. And I'm not sure there are that many, statistically or practically speaking. Sometimes when the ATF finds out about them they let them run illegally for a year or two to make a better court case. One case I am familiar with happened in Arizona where an FFL was selling guns to the Hells Angels. The ATF knew about this and let it continue, hoping not only to get him but some MCs as well. In the meantime, hundreds of guns were transferred to prohibited people--including converted full-auto guns--with the ATF's full knowledge. While they eventually arrested the FFL, hundreds of guns, including machine guns, are now out on the street, probably all in the hands of criminals. The ATF let that happen, on purpose.
b) stolen guns. What do you propose there, make it illegal to steal a gun? It already is. Make it illegal to break into a house? It already is. Make it a misdemeanor or felony for a gun owner to not own a specified security-level gun safe and put him in prison if he doesn't?
c) preventing mentally unstable people from owning guns. What do you propose there, abolishing HIPAA and making every psychologist and psychiatrist report their patients to the .gov? If a teacher thinks one of his/her students is depressed about bad grades or girls, report him/her to the .gov? Report anyone who is too anti-religion or too pro-religion to the .gov? Report anyone who criticizes our government to the .gov? And then change the law to state that anyone who anyone else thinks is suspect becomes a prohibited person?
d) universal background checks. Michigan, New York, Illinois, and Washington D.C. all require handgun registration and/or background checks. I'm sure there are others that do as well, these are the ones I know about. Where is there high crime with handguns? Michigan, New York, Illinois, and Washington D.C. UBCs do nothing to stop gun crime because criminals generally don't use legal guns to begin with. All it does is create a database of known firearms owners with types and quantities of firearms. That could never be misused by the .gov, could it?
Let's start with these four things that you have identified as problems. Propose solid solutions and let's see what they are.