I don't know the exact requirements but I believe that the requirements for the service entrance conductors or cable are different upstream of the disconnect compared with downstream. So by moving the service disconnect location from the pole to a building, you may need to run different kinds or sizes of conductors overhead or underground.
There is no overcurrent protection other than (probably) the pole transformer primary fuse for the wiring including the service conductors upstream of your service disconnect.
No changes would be needed for the overhead lines from the yard pole to the other buildings, now fed from a breaker in the main building's panel.
Must the service disconnect reside immediately after (below) the meter?
Here, no, but the reason this question pops up is that, in a typical home installation with the meter on an outside wall and the panel on an inside wall (in the basement?), there is a limit to the length of the pre-disconnect service cable in the house (again, no overcurrent protection other than back at the pole transformer) connecting the two.