Usually the phone lines pass through small holes in the studs in bundles often at angles around obstacles and can be hard to pull out, much less pull a larger cable through. You can try one in an area that won't matter if it doesn't work. Other than from adjoining rooms it usually is a chore.
Don't know about the fire code question, depends on the locality, but holes in the studs would be of more concern than the joists I would expect.
I've pulled a lot of cable through existing residential construction and the only time that it is reasonably easy is if it is a single story and does not have a flat roof, the larger the attic space the better. Commercial jobs are usually easier as many of the buildings have conduits where you can use a fish tape.
And for the cable runs that you can do, use CAT5e cable so you can achieve gigabit speeds.