Wood Floor on Concrete
as to the doors ,after you install the floor ,cut the bottom off the door.good luck
There vare some fine engineered products that can be glued successfully with the right professionals. Those would include Lauzon, Mirage and Margaratelli that can be refinished almost as many times as a 3/4" solid. I'm not sure why carpetman has suggested Bruce as I don't know many true hardwood retailers that even carry Bruce for obvious reasons.
I don't understand this comment either..." puting a plywood floor over cement is asking for moisture problems" Could be a regional problem. With hardwood flooring there are so many options available today, however it's not likely you'll find someone with the knowledge at a carpet store or the big box stores.
More On Hardwood Flooring Choices
go to the national wood flooring association website for more info.
Another company that makes a great product is MERCIER. They offer a 5/16 glue down product with a gorgeous finish.
What you are looking to do is very much like most gym floors are installed. Nail down flooring over a concrete slab. Many gym floors are 33/32" sugar maple but some schools save money by using 3/4" maple.
One thing I have done is to put down carpet pad or use the existing carpet pad and lay a layer of half inch plywood the same way as the walls. Stagger the plywood so the seams don't match up. If you use existing carpet pad you can leave the tack strips in place if there are any and go right over them.
I then put a second layer of plywood 45 degrees to the walls and screw the two layers together with 1 inch screws. By laying the second layer at the diagonal you insure that the seams between the two layers can't possibly match up and you are essentially making one sheet of plywood the size of your room.
This is how many gym floors are installed only they use rubber pads instead of carpet padding for the bottom. The height issue is one you will need to deal with if you go this route but it is possible. That may mean cutting the doors and door jambs so the flooring can slide underneath.