Yes, there are electric in floor heating elements which are quite thin and can in installed on top of plywood sheathing.

Yes. There are a number of considerations to take into account, but it is all do-able. Your best bet is to contact a local rep for any of the major companies and talk to them about the details. We've used NuHeat with great success.

Things to consider:
- Concrete slab? Will make a difference in how it is installed to assure heat does not sink into the concrete.
- Fixture locations. Once it's in, you can't really add or move anything that requires a new hole in the floor, so plan the holes out in advance.
- Electrical requirements. You will need at least one new dedicated circuit.
- Sizing. Are you looking to heat the room or just not freeze your feet?

We installed in 2 baths. Nice and toasty warm toes all winter.
:)