Actually, many of your suggestions are already in place. Aside from certain loophole situations in red states, every gun purchaser must submit to an FBI background check. In extremely restrictive states like California, gun owners must submit to a background check at the beginning of the sale, and then a mandatory 10-day waiting period, before the sale can be concluded.
Unfortunately, these restrictions have not reduced gun violence caused by street gangs and other criminal organizations -- in fact, the problem is getting worse.
Also, the mandatory training and re-certification requirement proposals are not going to fly at all. Drunk driving and texting while driving continue to be massive problems in the U.S. (and other countries) yet mandatory training isn't going to do anything.
I believe the core issue is that we have a culture problem -- we don't respect laws, authority, traditional ethical principles, etc. We have a Lord of the Flies situation, and no amount of gun control will stop gun violence, unless there was a way to completely eliminate all guns in the U.S. (I think it's impossible).