This is what I have observed about mine fields. Units on foot moving through a minefield using "quick" get blown up. If they move through using "hunt" or "slow" they are less likely to trigger mines, but it can still happen. Contrary to what is implied by the manual, engineers seem to have no special abilities in their ability to find and mark mines, and seem just as likely to be blown up as other foot soldiers. I have yet to see an opportunity for engineers to "mark mines" (the option is never available).
If someone has found a better way to deal with mine fields, please let me know.
I believe the way engineer units interact with minefields is one area of the game that still needs some improvement.