Does the market handle mushrooms of different types

Yes: loose and packaged white as well as crimini mushrooms. Loose and packaged portabella heads. Loose fresh shitake and button mushrooms.

Definitely yes. See Sandra's reply.

Yes they do. As you walk in the store, farthest aisle on your right. They're on the left side of that aisle, starting at the front of the aisle on your left. They usually have a number of varieties.

I don't think so. Mostly fruits and vegetables.