LaPierre is a hysterical moron. It's not worth paying attention to what he has to say, and it's not worth arguing with people who take him seriously, because they lack (or voluntarily surrendered) the ability to reason logically and surrendered themselves to emotions.
I would like to point out, though, that the law prohibiting mentally ill from possessing firearms is already in place. The point that NRA keeps raising is that the checks are done on federal level, while the databases are compiled on state level in a very ad-hoc and inconsistent way. Basically, each state decides what goes on the list and when it gets to the feds. So the effect is that we have a law on the books that is supposedly helpful (I'm not familiar enough with the topic to judge whether it actually is or not, though I suspect it would be with the proper criteria), but the implementation of which is hindered in practice. It stands to reason that either the law needs to be repealed, or it needs to be fixed to do what it's meant to be doing.
Regarding allowing people to buy guns after having convictions - I don't see a problem with this in principle. Rights are rights, and the right to keep and bear arms is there on the list alongside the right to free speech or to privacy and protection against warrantless searches. We don't refuse the latter to criminals after they have served their sentence, why should we refuse the former? (BTW, the right to vote is also one thing that is unfairly denied to such people, and that so many states still do it is a travesty).
The bigger problem there is the justice system that's designed to be punitive in nature rather than corrective or deterrent - we release criminals on the streets knowing full well that they're still sociopathic, and we put people in prison for years for crimes that don't harm anyone, or that they wouldn't repeat anyway because they already understand the error of what they did (and after they spend those years in prison, they often turn into sociopaths). Fix that, so that sociopaths remain isolated so long as they remain a danger to society, and gun rights for felons becomes a non-issue.
As far as terrorist attacks in malls go, it would seem that actual terrorists don't have a problem sourcing weapons for them even in countries where gun laws are fairly tight (like, well, France). In any case, in US, even if you were to make them all illegal overnight, you'd end up with all those millions of guns ending up on the black market, still readily available for those who intend to use them for some nefarious purpose. It would take not years, but literally decades for the circulation to scale down - a time scale that doesn't really mesh with the "here and now" nature of the terrorist threat.
And, of course, terrorist attacks are usually not gun massacres. Explosives were, and remain, a best and most reliable way to wreck havoc on a large number of people at once for maximum shock value. In Nairobi mall attack, it took at least four (almost certainly more, initially they reported closer to a dozen attackers; 4 is how many they have arrested in the aftermath) guys with AKs, engaged for several hours, to kill 67 people. In Moscow metro bombings, it took two women with suicide belts a few seconds to kill 40. Much easier logistics, too - no training necessary for the bombers, and explosives are homemade stuff with nails and scraps of metal for a shrapnel load. Pretty much the only reason for them to use guns is when they want to fight off police (in e.g. hostage scenarios like Beslan, or simply when they want to make a statement of how inept the government forces are by holding them back, like in Nairobi).