Materially speaking, the station is owned by the Earth Alliance Central Government. The other governments chipped in to get the thing built, but that was in exchange for a seat on the council, not their name in the deed. Indeed, they weren't the ones that funded the station's day-to-day operations, that was Earthdome.
So yes, they basically stole it from the Earth Alliance. But that's always true when any body of people declare political independence, the "steal" the ground under their feet along with the buildings, infrastructure and any other assets that happen to be there. But as I said though, no government, not even Earth has claim on the territory where the station orbits.
Not sure who owns the jumpgate. Probably Earth, but again, not much they can do about that. Those things are expensive, but it's a given that they'll be around much longer than any political change. Even attempting to take one of them out makes about as much sense as tearing up the roads and bridges around a city-state out of spite.
The relationship between the station and EF was a little complex. Yeah, the station commander and the command staff were all EF officers, but the Earthforce military wasn't technically in command. Sinclair and Sheridan were military governors acting on the direct authority of the EA President and overseen by the Earth Senate oversight committee. Hence the pissing match that ensued in 'Voice in the Wilderness' when Captain Pierce tried throwing his weight around. Although he outranked Sinclair, he had no authority over the station or it's personnel. It's also why when Sinclair was recalled to Earth, it wasn't some General or Admiral, it was Clark that gave the order.