Given how last season they called the plan to get back to Earth the Exodus Plan (or something with Exodus in it), I'm pretty sure that when Kane mentioned the Exodus Charter he was referring to a rules they had formulated to be in effect once they returned to Earth, and "execute anyone who breaks the law" probably isn't one of them.
And keeping up the the Kane defense, getting ticked off over who has access to the guns is perfectly reasonable. The have limited resources so they need to use them to best effect, and so should the guns be going to trained security personnel or to just any yahoo?
Also on the security: Kane caught some flak for not sending out patrols to look for the lost lambs, but making sure your base of operations is secured before going out and potentially looking for trouble just makes sense. What's the point of bringing people in if all you're doing is putting them all in one convenient location to be wiped out because you didn't make sure that one location was safe?
Are the Mount Weather folks stealing the Grounders' blood or using their bodies as a sort of dialysis machine? More importantly, why haven't they been trying to figure out how to adapt their bodies to the environment? I mean, sure, they seem to be comfy cozy in the facility, but they have to know they have a critical strategic weakness that could doom them in the long run. That's my theory as to why the 48 aren't in the cages: unlike the Grounders, they're from a technologically and culturally sophisticated society, plus there's only 48 of them and they're young, so it should be possible to assimilate them into the Mt. Weather society where they will eventually disseminate their ability to withstand the outside environment throughout the population.