Your electrician has already shown he doesn't really understand what is required. Your biggest problems here are first time, and second convincing your contractor to do some things he will think are a waste of time and money. RG6 is fine for cable, satellite, and over the air TV distribution(OTA). As has already been said, forget HDMI over RG6.
In a nutshell,
1) do double runs of RG6 from your central closet to each terminal point (star topology) (remember, cable is cheap, cheap);
2) forget CAT5 and forget CAT5e, which is over 10 years old and has much greater bandwidth that CAT 5. Do double runs of CAT 6 (gigabit Ethernet) in the same star topology as the RG6. Cable is cheap, http://www.monoprice.com
charges around $155 for a 1,000 foot roll of in-wall rated CAT6.
If you do these two things, you preserve your options. (You will also need cable feeds to your central closet for the outside source signals for cable, satellite dish(s), over the air TV antenna and the cable/dsl modem (for high speed internet)). I'd also consider "whole house surge suppression."
To expand on the above,
1. RG6 is what you need for conventional runs from the cable or satellite company. If your contractor fights you on dual runs, point out that satellite dish runs already require dual runs due to the bandwidth requirements of all the transponders.
2. Cable is cheap. Also, if you want over the air local tv reception, that will require additional RG6 runs as well. Conventional over the air HD broadcasts can still offer the best HD signals, second only to bluray. If your builder tells you you don't need it because cable and satellite rebroadcast the local stations, tell him to run dual RG6 distribution lines anyway. Unless you have fiber to your home; e.g., FIOS, you will discover the cable and satellite companies are delivering softer and softer "HD" signals because they can cram in more channels by lowering the channel bandwidth (bit rate) which lowers the HD content delivery quality). Finally, cable and satellite companies are always fighting with the local broadcasters over licensing fees which result in blackouts and are also prone to imposing blackouts for other unfathomable reasons. Bandwidth, flexibility and redundancy are the central ideas.
3. Currently, if the cable source is DISH, they have their Hopper/Joey system that allows 1 big receiver/digital video recorder box to distribute the signal to a tv connected directly to itself and to 3 other tvs through a much smaller box. Multiple big boxes can be networked for more than 4 sites. Competitors will undoubtedly offer similar solutions, but not for a while. These receivers all have RF remotes and will work through walls.
4. You won't get HDMI 1080p signals out of RG6 unless you buy a pair of RG6 to hdmi converter boxes for each point to point link in your network; further, they are currently expensive ($100-$300 a pair). Long HDMI cable runs become expensive and impractical (repeaters or fiber optic cable).
5. You will get HDMI out of your satellite and cable boxes and from them you run an HDMI cable to your TV or to your Audio Video Receiver and from it to your TV.
6. I'd start out with one good bluray player. For example, an Oppo digital bdp-103 (a bdp-105 if you have audio/videophile tendencies and the budget), connected to your AVR (audio video receiver) in your media room by HDMI cable for HD audio and directly to your TV for the video. After that, unless you watch movies in bed, I'd buy secondary bluray players; they've become quite inexpensive (as low as $30).
7. I'd forget the idea of martix switching over RG6. HDMI over ethernet is much cheaper. For example, while not matrix but broadcast, a 1x8 (8 room) HDMI Amplifier Splitter over CAT5e/CAT6 Complete Solution Kit for cable runs upto 164ft is $271 at http://www.monoprice.com
. Point-to-point switching, if needed, is not that much more. If the wire is in place, you can always swap out the electronics.
8. NO CAT5 Ethernet and NO CAT5e Ethernet cable. Insist on CAT 6 (gigabit ethernet) for your ethernet backbone cableling. If your contractor fights you on this, he is ignorant. Don't let him talk you out of it. If you have to, go buy a box or two of CAT6 yourself and give it to him. If you have time, it is much cheaper to buy over the net. As I said above, a 1,000 ft box of in-wall rated CAT6 cable is $155 at Monoprice.com. As with the RG6, I'd run dual backbone CAT6 cables, especially if you are really considering matrix switching audio/video over cable.
9. Depending on your budget and available time, you could also run conduit. For example, visualize 1 to 2 inch in-wall pvc pipe (cheap) from basement to attic with junction boxs in each floor; drop at least 3 or 4 pipes so you can reach any room/closet from any other room. You don't have to put anything in them now. Futureproof your house unless your are not planning on staying. Will you need it? Who knows. I did it in a house I built in 1981 and it didn't take long (10 years) for the 1" (inside diameter) pipe I used to fill up.
10. Back to matrix switching. Run your CAT6 IP network in a star topology with your gigabit ethernet switch as the hub in your central closet. You will probably need at least a 16 port, non-blocking, gigabit matrix switch; but these are commonly available for under $100 (of course you can pay a lot more). You can/will soon be able to do all the wired audio/video streaming/switching/distribution you want over your IP network. For now, also run your second CAT6 HDMI audio/video network using the same topology. Remember, cable is cheap, cheap, cheap and it takes essentially the same time to run two cables as one. Did I say cable is cheap?
11. Finally, more and more devices also have wireless ethernet capability. Even speaker wires are going wireless (no standards yet). This can eliminate a lot of device to device cables. Audio and 1080p video streaming over wireless ip is pretty good, as long as your wireless access point(s) can handle your bandwidth demands. This is true whether your source is over the net or from your home network file server (most of which will require a wired connection to a wireless access point). Why do you need in-house cable anyway? Wired connections are far more reliable and can handle far more bandwidth. Wireless may be the future but it cannot yet supplant wire totally. Even if it could, do you want to be continuously bombarding yourself with many more, admittedly low power, high frequency wave transmissions?
Much is left unexplained and unsaid and what has been said mainly reflects my biases. Good luck.