I'm pretty good at physics, and i know the film quite well, let me have a go at answering the questions:
1. a=rw^2 If I knew the station's radius I would be able to tell you whether it is spinning fast enough, the rate of rotation, in radians per second must be high enough that when multiplied by the radius, in metres, it gives a value for a (the centripetal/centrifugal acceleration) of 9.81 metres per second per second. As a note the spinning section in the "head" of the Discovery would probably not be able to generate significant artifical gravity , given it's radius it would need to rotate fast enough that the crew would struggle to walk in straight lines due to coriolis forces and would feel quite sick due to the much weaker gravity field at their heads than at their feet.
2. This could be done, but it would be a tricky maneuvre, the shuttle would need a to line up with it's nose-tail axis along the axis of the stations rotation, then make a long burn with it's roll thrusters until it was rotating in the same direction with an angular velocity (w) to match the stations. The shuttle crew would not need to worry about generating their own artificial gravity like this because the shuttle is so much narower than the radius of the ring of the station. You are right in saying that a tunnel which did not rotate with the station might be easier, but the method shown is indeed possible.
3.If you had enough air in storage, and your air locks were properly tight, and your pump were good enough then this is possible. Think of it as a giant airlock. The shuttle comes in, you seal the outer door, make sure the seal is very good, then you let air out of pressuried tanks and into the airlock, then open the inner door, let pasengers disembark and cargo be unloaded, leave the airlock full of air until the shuttle has been refilled with passengers an cargo going the other way, seal the inner door, suck the air out (as long as your pumps are good you should be able to get all but a few hundred grams of air back into the tanks), then open the outer door and let the shuttle leave. This would take quite a lot of energy to do all the pumping, and it would be rather slow, but it could be done with such small percentage losses of air (a few hundred grams on every cycle because not all the air could be sucked back into the tanks) that you could operate it for quite a long time before you needed your tanks refilling. The tiny losses are because any pump used to suck the air out of a space will eventually reach a point where the pressure within the volume it is emptying is too low for it to continue sucking, but the presure where this occurs for most pumps is low enough for these pruposes. An air tight tunnel connecting the shuttle (or moon ship)'s airlock to one on the base would be more efficient but this giant airlock method is still possible. Airlocks on the ISS definitely do recycle air by sucking it out of the airlock before the outer door is opened.
2001: A Space Odyssey is certainly one of the most scientifically accurate sci-fi films I know of. One notable error though is the lack of heat radiators on the Discovery, they would be needed because it is clearly a nuclear powered spacecraft and without them the ship would overheat. Although space is cold it is hard for a spacecraft to stay cool, because the only way they can lose heat is by radiation (where vehicles within an atmosphere can lose heat by conductiona dn convection effects too). Power output from radiation follows a law of the form P=kAT^4, K is a constant, A is the surface area of the radiator, T is the radiator's temperature, P is the rate of heat loss. If the area is too small the ship would equalize it's rate of heat gain (from sunlight, onboard systems and especially the reactors and engines) with it's rate of heat loss at an uncomfortable, or more likely lethal, temperature, so you need to give it radiators with a large area to keep the temperature cool enough for your crew and other systems. Some of the original concepts showed it to have large panels for this, but these were not shown in the film because some of the audience would have probably though they were wings, which a deep space vessel does not need.
Edited by rp88, 26 April 2015 - 10:50 AM.
Back on this site, for a while anyway, been so busy the last year.
My systems:2 laptops, intel i3 processors, windows 8.1 installed on the hard-drive and linux mint 17.3 MATE installed to USB