I know you likely mean well, but you might want to think through some of your ideas.
The problem with mandatory mental health screenings is, who is doing the screening? The medical and mental heath professions have always been anti. If you allow the AMA to declare people unfit, you will see an increase in numbers of people with mental issues that you would never have imagined.
Federal registration is a horrible idea and the first necessary step towards confiscation.
Mandatory firearms training doesn't fly with me. There are a ton of people in this world that will never be comfortable with firearms. Don't force them to try. Make it an elective if you want.
There really is no reason for licensing at all. The Constitution pretty much covers it. It is just a way for a state to generate revenue while pushing a political agenda. And PP doesn't fly with me either. If a man commits a crime and serves his sentence, shouldn't he be able to protect his life and the lives of his family members? It's a basic human right.
If he is a danger to society, wtf did you let him out of prison for in the first place?