If a a Military Operations Area (MOA) is designated from 10,000 to 18,000 feet, can I fly under the airspace without any permission?
Yes you can. The MOA floor, 10000ft in this case, is the lower limit of any restrictions a MOA imposes. Otherwise there would be no point in assigning a floor to MOA (or any other airspace for that matter) See FAA AIM Section 4. Special Use Airspace