How to decide. Well as the 2nd Amendment gives no qualifiers and simply states shall not be infringed. I would posit, that if a person is considered safe enough to walk the streets. Then they should be able to exercise their right to carry. If they are too mentally unstable or criminally minded to be allowed to carry then they should be locked up, either at a mental health treatment facility for the mentally unstable, or prison for the criminals. Psychological treatment alone is not grounds for stripping a right. It must be those adjudicated in a legal proceeding to be unsafe. No more allowing the VA to strip veterans of their rights simply because they aren't good at managing their money, leading the VA to appoint a fiscal manager over their funds.
At what age? The age when one is no longer legally considered a child in the US, age 18, plain and simple, but I'd allow a waiver down to 17 for anyone who has enlisted in the military. Not that I trust them that much more (though after 20 years in the Army I do), but if they can be allowed to put their lives on the line, then they damn well can carry a firearm at home if they choose.
Should they be allowed in all buildings? No not all buildings, some buildings do have a need for higher security. In my state those are identified as jails, prisons, courts, and secure mental facilities. And of course the secure areas of airports, post offices and anywhere else the Feds have deemed off limits (I do disagree with many "federal facilities" that get the protection, but congress just made an overreaching blanket protection). I would like to see the Post office ban removed and a case is moving through the courts that is likely to do just that, so far the initial ruling is against the Post Office. And in my state those previously mentioned secure facilities are required to provide a location to securely store your weapons as well as a clear demarcation that you are in fact crossing into the secure area. I do note that the courts in the state (Utah) ignore that law, saying they set their own rules and they will just charge anyone who attempts to pass beyond the security checkpoints with contempt of court, yet they provide no storage areas.
What about schools you ask. Well thanks to the Federal Gun Free School Zone Act, for most citizens schools in the k-12 range are off limits, and so are colleges and Universities. But the GFSZ act does allow for the state to permit carry. And so with a Utah Concealed Firearm Permit, we can ignore the Federal law and carry into our schools, and we have been able to for nearly two decades now. Similarly our Public colleges and Universities are open to any with a CFP.
At any given time an unknown number of Teachers, administrators and visiting parents are carrying within our k-12 schools. Good luck trying a Sandyhook when our teachers can do more than just be the first victim. Not all do carry, I'd guess most do not, but enough educators have attended free CFP classes to indicate that few if any schools are ever gun free. So other than the four off limits places and the federal restricted areas, everywhere else should be open. It's not quite that way as there are a couple ways that churches can also prohibit carry (without any requirement to provide secure storage). Otherwise no "NO GUNS" sign posted in this state has any strength of law and that's how it should be. You can ask me to leave if you don't like my firearm (assuming you even know I have it since I usually conceal) and if I refuse then you can trespass me and prosecute for trespassing, but not for having a firearm.
Planes I would allow, just require passengers to declare that they have frangible rounds in the weapon. Let's see a terrorist try and hijack a plane when a sizable fraction of the passengers are armed. Train/bus/etc absolutely. Unlike a plain that has special considerations regarding the potential of a weapon being discharged (decompression). There is nothing special about those. Except that they have the potential of putting you in close proximity with low life scum who wish to do you or others harm.
Now others will come back and claim that if we were to allow such, we would have planes falling from the sky, and blood running in the streets. Except that every time relaxed firearm laws are proposed those doomsday predictions are trotted out, but such predictions never come true. Instead as firearms laws have been relaxed across the country over the last couple decades all violent crimes have decreased.
In fact studies tracking those who have obtained concealed carry permits show that such citizens are 4 times less likely than police officers to commit crimes. And in defensive gun use after defensive gun use, innocent bystanders are not getting hit by stray bullets. Why not? Because we who carry know we have to pick our shots, we will be criminally and civilly liable for every bullet we fire.
The police on the other hand are protected by qualified immunity and the blue wall. So we see the police in times square injure nine bystanders taking down a single armed criminal, who didn't fire a shot. Or we have the Police in LA shooting up a truck, of the wrong color and make with two female occupants, because its within a few blocks of a potential victim/target of their cop-killing ex-cop suspect who was a male. The city just paid out a big settlement to those two ladies who luckily survived, but those officers are back at work.
Now find me an example of a citizen shooting gone bad resulting in innocent bystanders being injured. I won't say they don't happen but they are very rare. And Trayvon was not such an event. Right or wrong, one shot was fired and it did its intended job, no innocent bystanders were injured.
Instead we find events like we saw two days before the Sandyhook shooting. At the the Clackamas (sp?) mall shooting, where an armed citizen drew down on the shooter, who was clearing a jam in his AR-15 but contrary to all the doomsday predictions of shootout at the OK Corral, he did not fire randomly, in fact he choose not to fire because there were innocents beyond the shooter, so the citizen withdrew moving his family to better cover and safety, ready to defend but not firing when he knew he couldn't do so safely. Now it can't be proved this stopped the shooting. But he did make eye contact with the shooter, who then ran off and put a bullet from one of his two handguns into his own brain ending the event. Maybe he quit and ran because he couldn't clear the jam, but he did have two handguns and could have kept on killing. But he didn't, it is suspected that once the shooter realized there was armed opposition he did what most such shooters did, killed himself. We saw this at Sandy Hook and have seen it time and again. They shoot and kill until they get the first hint of opposition and then they either kill themselves or they surrender.
Sorry about the Epistle. Hope I gave you something to consider.