Am I correct in understanding the rules for minefield that it is fine to start standing on top of them, it is only if you end your movement within 6" they go off? This was a concern I had while trying to squeeze in minefields and my regular deployment troops at the same spot.

Correct. There is no effect for starting your turn inside a minefield.

Well, if you are standing around and actually see the mines deployed around you, if you don't complain, you can at least try to memorize where they put them..

What about Units with the "Fast" ability? Can they cross a minefield without any effect when they take a march move action (= 18" = outside the minefield diameter)?