Am I correct in my assumption that most stationary computers with
> broadband stay in the same place.
Probably, mostly, but maybe not forever. Don't forget wireless.
and they are almost always on the same IP address as well?
That very much depends on the ISP. Mine (DSL) does, but a client's (DSL through a different ISP) changes IP every few days. Many cable systems change IP number periodically. Sometimes getting a fixed IP is an extra-cost option. But you can't count on it as a default.