Placement of electronic appliances in close proximity to water is generally unwise. It is usually standard - and more convenient - to mount a microwave over your range or cooktop.
Yes, but not because it is a microwave. Any electrical appliance under or around your sink should be on a ground fault interrupter. Water plus electricity may yield electrocution.
Put some distance between the sink and the mircowave and it should be okay. People have their microwave over their stove. Cooking on the stove produces steam with little problems.
