Any of those would have made him a prohibited person from buying or possessing a firearm legally or possibly had him locked up which would have prevented that shooting by him.
Not exactly.
First, you'd need a conviction. (or guilty plea)
THEN, you'd also need that it not be a juvenile conviction and that the records not be sealed or expunged when he turned 18.
A background check doesn't check expunged records, nor can it look into sealed court record, either. It might find sealed records and cause a delay or it might not, I don't know the system well enough to know if it could.
NICS is not the indepth kind of investigation done for a security clearance. Some want it to be, but currently, it isn't.
And, even if it was, its not a guarantee that any individual won't commit murder or mass murder. We've seen it happen, both in Texas and Florida in recent years.
And, even if he had been convicted and become a prohibited person, that is no guarantee that the shooting would have been prevented. Only that he wouldn't have been able to get the gun through legal channels.
Nor would having him be locked up be positive prevention, unless he stayed locked up. These things might have changed his mind, and therefore "prevented" the shooting, but
might have is not
would have.
They might only delay the killer. They might only have prevented him from going on his killing spree on that particular Valentine's Day.
Claiming that had these things happened he would have been stopped is intellectually dishonest, and wishful thinking, at best.