When parked at the gate that requires a push-back onto the taxiway, from which ATC station is the push-back clearance requested? Is this the ground or apron station, or is that still departure clearance delivery station (as they better know the slot times, etc.)?
At controlled aerodromes, the aerodrome control tower is responsible for separating aircraft on the maneuvering area (from other aircraft, and from obstacles, vehicles etc.). If a pushback is done onto a taxiway which is a part of the maneuvering area, the aerodrome control tower must approve this first. At large aerodromes, there are different tower working positions, operating on different frequencies and with different callsigns. So the "which exact ATC station" question really varies from airport to airport. You might talk to "xx tower", "xx ground" or something else entirely.
Note that at many airports, the gate area (apron) is NOT a part of the maneuvering area. Technically speaking, ATC is not responsible for separating aircraft outside of the maneuvering area. Instead, pilots, tug drivers, vehicles and other persons on the apron follow different priority rules when a pushback is required (traffic laws normally state that vehicles shall yield to aircraft; the rules of the air have rules regarding aircraft right-of-way). However, some major airports have designated positions for coordinating this effort, which - from the pilot perspective - looks just like ATC. They operate on a VHF frequency, and typical callsigns are "xx apron" or "xx ramp". Note, however, that such stations do NOT provide air traffic control - technically, they provide advisory service, and they are not responsible if a collision occurs, because ATC separation is not provided outside the maneuvering area. Persons staffing such positions generally do not have to be fully qualified air traffic controllers (this will vary from country to country), and are sometimes employed directly by the airport, instead of an air navigation service provider.
In summary: if the pushback occurs on the maneuvering area, the aerodrome control tower has to approve it. If not, no ATC approval is required, but there might still be other stations coordinating pushback.
It depends on the airport.
I don't have a statistics, but from what I've seen, usually it is the ground station, as they coordinate the movement of airplanes on the ground, and they have an idea if the runway lineup queue is long, or another plane is taxing behind.
In non-busy airports (the ones which only get a few flights per day) the same guy might be the clearance delivery, ground and tower at the same time. So the pilots are basically talking to the same person.
KJFK is one exception where push back is coordinated by "ramp", as you can hear for yourself in this ATC recording with Air France 011 Heavy who was apparently was not very familiar with that airport's operations.
Most likely either ground control or if there is a designated frequency for a secure apron movement, that would be used. Check the A/FD entry for the airport or ATIS for further information.