Can a player build airfields or ports in a territory controlled by a friendly player. I know you can not build factories, but for example can the American player build airfields in territories controlled by the British, as any friendly player is able to use airfield unlike a factory.
No. Rules, pg. 24:
Air bases can be built on any controlled territory or island.
("controlled" not "friendly")
and pg. 22:
Move the newly purchased units from the mobilization zone on the game board to eligible spaces you have controlled since the start of your turn
Place new facilities in any territory that you have controlled since the start of your turn.