At NASA, the type of vehicles you are describing go by the name Orbital Transfer Vehicles (OTVs). While none are currently funded or planned for, there were studies underway as recently as the 80s.
Upper stages are sometime considered "space tugs" but I don't think that is a good description as they are only used once and are really just, well, an upper ...
An on-orbit rescue tug isn't likely to be viable in the near future.
Automated docking is typically only done at the ISS and requires extra hardware, both the docking port and the radar systems to align vessels. This is a non-trivial amount of additional mass and power required just to allow a space tug to have a chance of rescuing your satellite ...
I can't find anything on this online. It would seem to be a useful technique as it would lower the delta-v for the vehicle that lifts the payload from the ground to the rendezvous point relative to fully making orbit. (NOTE: Yes, you'd have to get extra fuel to the tug to perform this manoeuver but that's potentially a smaller problem.)
This is the opposite ...
Is this theoretically correct? Is this practically feasible?
I get the same result using the rocket equation, yes.
The tug described can move a 47 ton payload through that Hohmann transfer; it will then be completely out of fuel and unable to maneuver anywhere else.
The Orbital Sciences entry in the CRS program is the Cygnus vehicle. It has two components a cargo module and an orbital module. While not officially offered as a tug, it functions as a tug to bring the cargo module to the station.
While the Cygnus payload module is not that big, it seems likely that Orbital could provide such functionality, should the ...
Googling around, I couldn't find any mention of a space tug program being officially announced by NASA.
A reusable space tug was studied in 1969 as part of NASA's Space Transportation System, but went unfunded, as did every other component of that system except the Space Shuttle.