I also suspect that the mag limit will be considered constitutional and I doubt it would be challenged specifically. It is a restriction not a denial of rights. Making many commonly available firearms illegal is another matter and that will be the big test if that in fact happens.
The mental health issue needs to be addressed, but it is a matter of degree in my opinion. Requiring for example a mental health examination prior to the purchase of a firearm would be restrictive of the right. Requiring you to provide personal references attesting to your sound character are another.
Requiring effectively all firearm transfers to go through a FFL is almost a given if a new law is passed.
Executive Order (EO) powers are not clearly defined. But I think it is clear that the President has the authority to restrict specific imports.
Strengthening Federal firearm laws without legislative action is debatable. But "smaller" revisions to Form 4473 and BATFE authority are likely.
The UN Small Arms Treaty is another concern but with Sandy Hook, ratification becomes a possibility at this point.
1. Regarding the mg limit, without confiscation and/or ex post facto, it would be useless, and also cause the same problem NY has with their mag limit, since date of mfg isn't stamped on mags anymore post AWB sunset, there is no way to prove it isn't grandfathered. Given the dramatic increase in number of modern sporting rifles in use with standard 20-30rnd magazines, and the dramatic rise in the maker movement, any magazine restriction will likely simply create a sound black/grey (since date can't be proven) market for modified magazines, not to mention production of modifiable magazines. Think the way designer drugs skirt the law, and give the incentive to more people with more money.
2. Regarding mental health of any kind (including references) not not no, but HELL no, it would've ext to useless in stopping instances like this, create a massive grey market in "straw references" or "one hand washes another" and create an opening for official adjudication and the associated slippery slope. It won't prevent how the crazies get weapons, and simply create yet another hoop for legal purchases by law abiding folks.
3. All transfers through an FFL eliminates the last impediment to national registration, as it allows the same type of EO or non-legislative regulation that they threw at the border states for long gun registration. Universal registration is ALWAYS (in every country that has done it) followed by restriction or elimination. It's no ones business how many firearms I own, jet like its no ones business how many laptops I own. Also, this wouldn't prevent any crimes, illegal transfers would still occur, straw purchases would still occur, and all it does is create a slippery slope opening.
We all need to face the cold truth, the only way to stop someone intent on killing others and going through with it is to stop them as quickly as possible, no laws will prevent it, only citizens and LEOs (when possible) defending themselves and others.