may i put an hdtv over a fireplace?

Pretty good rcprince.
I think it would be OK with a gas stove as long as there are no great heat problems. I would have it built into a sealed enclosure though for aesthetics, and to keep out smoke and lessen the heat. You could also have this enclosure somehow cover the screen when not in use.

Maybe you could just project a live video feed - people will think it's a mirror.

I've got better idea:
It's OK to me to hang it above the fireplace if you're not going to use it as a fireplace. Instead you could use it as component compartment.
But MAKE SURE that EVERYBODY knows that this isn't a fire place anymore!!!!!!!

No. Unless you have bleachers in your living room. If not you will have stiff neck, everytime you sit down. As far as heat goes, how many days in a year is there a need for fire, unless you live in Alaska. Isolation material behind and under the flat screen should take care of that.