What you want is a matrix switcher with splitter support. This is very different from an AV receiver. My recommendation would be to get one that supports HDMI audio and then have your tv audio out run to an amp in that room. That would be the cleanest way to do it. Otherwise you are looking at spending over 8000 for an av receiver that also does matrix switching
Here is an example octavainc.com/HDMI%20matrix% ... 204x4.html
Questionst to ask yourself :
A: do you ever want two tvs to display the same source?
B: where will audio be decoded/amplified?
C: do you want 3d capability?
D: how far are your runs?
Here's why
A: a switch matrix allows you to route any input to any output, however it does not allow you to 'clone' an image to two outputs. For that you need a matrix switch with a splitter function as well. Again, more pricy....
B: it would be least expensive to pass audio in the HDMI signal and then decode it on the tv end. My guess is you have a. 5.1 audio system with each tv? If so give up the dream of having EVERYTHING in the basement or you will be running six speaker wires to every room, which will degrade the audio signal and cause you tons of future problems. Instead, I would suggest you send digital audio to the tv, then use the tv's optical out to pass that to a 5.1/ whatever point whatever decoder and amplifier.
C: think about future proofing. Do you want 4k resolution capability? Personally I hate 3d, but I ran 4k capable HDMI cable because a house lasts very long. I also ran cat6a in the walls instead of cat6 or cat5e.
D: HDMI is only good for so many feet. After that you should look into cat6a video distribution systems. Again it'll cost you more, but if you are running more than 40ft of HDMI in the wall you will either need amplifiers (in which case get a cat6a converter it'll be cheaper and better in the long term) or you'll not be able to do 4k