I can purchase a fighter and deploy it on a new AC.
Yes.
I can also purchase a fighter and deploy it on an old AC.
In LHTR, yes; in the box rules, no.
Im not sure if you can purchase a fighter and deploy it on an allies AC. (anyone?)
No.
Can I purchase an INF, RTL, AA, or ARM and deploy it in a new transport?
No.
Can I purchase an INF, RTL, AA, or ARM and deploy it in an old transport?
No.