What cable to run in new construction house

I'd run cat 6 everywhere, you'll be able to use it for purposes other than internet if you desire in the future. You can use it to get your HDMI to the TV as well. Don't know your budget, but look into HDBaseT stuff to distribute content to multiple TV's in the house

You should check with the builder. Most won't let you run your own cable.

It depends what you want. I originally ran 2 cat5e, 1 RG6 and 1 telephone wire to each location in my house, before I figured out that telephone wire is useless (wireless has replaced that), and back when cat6 was too expensive. I've since upgraded to cat6. I have a ton of RG6 in the attic for satellite, one run for cable, all of which goes to a central location. More RG6 goes to each room. Tons of cat5e/cat6 go from the central location to multiple rooms. I've run multiple cables for speakers (two to garage, two to the deck, and two in the attic for future expansion). If it was me, I'd also run conduit so you can add/replace later.

Pre-wire questions are better asked in the Home A/V Distribution forum a few down from here...

But in general - lots of ca5e/cat6 everywhere in addition to at least basic RG6 coverage. Read the Cocoontech Wiki for a lot of ideas as to where/why to run cables:

http://cocoontech.com/forums/files/file/64-wiring-your-new-house-101/

Also generally, don't run long HDMI cables in the walls for distribution. They are problematic at best, and you want to have a consistent wiring solution for distribution. But wiring from point to point in a room (for a projector or wall-mounted TV) is fine with HDMI runs.

Jeff