There’s nothing much that prevents you from lying when you fill out Form 4473 to purchase a firearm. The threat of spending 5 years in prison is not much of a deterrent if you’re planning to murder a bunch of people.
No, but there’s a database of felons and other people (such as some of the mentally ill) that the gun dealer no doubt checked. This guy wasn’t on it.
The Sandy Hook killer wasn’t on the list because his mother had bought him the guns. She tried to lock them up later, realizing that even she couldn’t deny that he was a homicidal maniac, and he killed her to get them.
If you don’t think it’s going to be a problem that this most recent killer bought his guns legally in his own name from a legitimate dealer who no doubt checked the criminal registries, you’re burying your head in the sand.