1) There's no simple way to check, but Legends Mode is where you'll find the information. You usually can't open Legends mid-game without retiring first (Dfhack has a way, not sure if it's working perfectly yet).
The old fashioned workaround is to make a copy of your save folder (or the the whole damn dwarf fortress folder, it's not big) then retire from that and check Legends.
2) Export of axes banned, means axes are banned from being sold to traders. Sell one and someone will be punished (if you have a Captain of the Guard / hammerer). Trade goods are filtered to not show banned goods by default (as far as I recall). So if you seem to be missing some bins, it might be because there's a banned item in it. You can turn off this filter, just don't accidentally sell the banned item.
Other mandates are construction orders, just make what they ask for and it'll automatically keep track.
Some make Demands too (like for impossible to make platinum beds, etc). They'll be upset not to get their stuff, but it's not against the law (I think).
Don't use the justice system (hammerer, guard squad) and nobody gets punished. The noble's mental health might take a hit, but don't worry about them. (That's in the current version at least, keep an eye on updates).