A genuine "geek house". One of the local variety had a nifty way of doing networking. They installed plastic rain gutter all over the house near the ceiling. Just drop the cables and wire into the rain gutter and pull out the ends as needed. No ugly wires on the floor, across halls, etc. It worked amazingly well.
It might be. All you need is a broadband modem, a DSL/cable ethernet router, and some wireless access points. Note that I said more than one access points. 6 others implies a large house with more than one floor. Getting reliable wireless coverage to the entire house may be difficult, especially between floors. So, you run CAT5 cable from the router to mulitple access points all over the house to insure decent coverage. One of the boxes can be a combination router and access point also known as a wireless router. The others are just access points (no router) but can be a wireless router setup as an access point.
I would run the CAT5 cables and not use wireless unless running the cable is deemed impossible. The laptops probably already have wireless capeability so those can connect with wireless. The desktops should get wired if possible.
Depends on the construction. I would start off with a LinkSys WRT54G and the LinkSys wireless G cards (if you need them). If one doesn't cover the the needed areas, get another then load Sveasoft Alchemy or Talisman firmware and do a WDS to connect them together. I just did the later for my brother who lives in a concrete construction home and it works great.