It will be dependent on the equipment you choose, particularly the control system and any matrix switching devices, as to what wires you will need to run between the basement and your upstairs equipment room and the wires running to each room.
If I understand your brief description correctly, I would run the basement video to the theater as a source input to an HDMI Matrix switch and then the remaining home runs would be from the theater to each room. Seems pretty cut and dry.
Personally I will have all of my WHA and theater equipment co-located in two Middle Atlantic racks. The only equipment in each room would be any in-ceiling speakers and / or a flat panel...that's it. Managed switches, digital phone system, etc. will all be in the basement racks / equipment room.