Will one router reach everywhere?

It depends on the construction, what the walls and floors are made of, and where you put the router.

If you put it in a corner on the first floor, the signal probably won't be very useable in the opposite corner on the third floor (but it may be). If you put the wireless in the middle of the second floor, it would probably cover the entire place.

You can plug the wireless router in without anything attached to it now and see how the signal is throughout the place based on where you put it, that would give you an idea if you can get by with just the router you have now.

The good thing is if you need to add an access point, having wires pulled already is a major plus.