The best way to answer these questions is just to design a very simple scenario. Sometime when you have nothing better to do, set up a simple scenario with some engineers(pioneers), some arty observers of different types, and mines. Then run through it.
You'll find that when moving forward (say with move or contact orders) the infantry units will generally discover the mine field before they hit it, though they will only stop if on contact orders. If they are running, advancing, etc they probably will not spot them prior to running in to them. Thats for AP mines. AT mines are much larger and were generally simply laid on top of the ground (obstacles of this type serve as a detterent to movement thru a certain area rather then to kill tanks). So they are much easier to spot and wont harm your infantry if they walk or run through them.
Spotting of mines by tanks, of course is much more difficult, particularly if they are buttoned up. Thats why its a good tactic to have your mine fields covered by indirect fire. That way you can call in fire, causing the enemy tanks to button up just prior to them hitting the field.
Mines of all types CAN be destroyed by arty fires though they have to be concentrated and of sufficient caliber (usually 150+). Otherwise you have to move pioneers up close and have them do it by hand or with satchel charges. This takes time however, usually 2-3 turns with a full squad, longer if the squad is under fire or has taken casualties.