I will be installing unfinished red oak hardwoods in adjoining rooms in phases that are likely to be 6 to 12 months apart. When installing in the second adjoining room at a later date, will it be necessary for me to completely refinish the first room to get the finish in the two rooms to match?

As long as you're using the same hardwood along with the same finish, it shouldn't be an issue. What we ended up doing was putting painters tape along the seam between two boards in each doorway that we were going to be finishing. That way, when we came back around to doing the hallways, we just put tape down in the other direction to protect the original finish. Peel it up, and voila!

It can be done, great care needed at the junction between the older finish and the new, to avoid overlap.
Personally, I would bring the second phase up to the number of coats applied to the first phase the apply a top coat on the whole floor.
I realize this will involve furniture movement, but it IS your home, afterall.

If topcoating the whole floor after coating the second floor with the same number of coats as the first, would any floor prep be required on the first floor that is 6 to 12 months old?

You will screen the entire floor, old and new; before applying the top coat.