The advice I'm providing you is specific to your situation and unit and is not necessarily helpful to others:
1) Do the wires from the compressor trigger the fan or visa versa? Yes & No. There should be a contactor that will trigger both the fan and the compressor at the same time.
2) If the wires are corroded too close to the point of entry on the compressor, is there any way to repair them or will I most likely have to change the compressor. If you have just enough of a lead on the wires, consider soldering on a pigtail to give you something to work with. If not, unfortunatley, the compressor will have to be replaced. I believe there is a small plastic or metal cover where the wires connect to the compressor, remove this if you haven't already and see if the bolt down to lugs where the leads can be removed from the lugs and splice in some new eyelet connections to the lugs.
3) What are the odds the compressor is bad (vs the relay or wires to the compressor keeping it from running)? 50/50 odds, you won't know anything until you get the wiring repaired.
4) Given the age of the unit, if the compressor is bad or the wires cannot be repaired (and thus the compressor needs to be replaced either way), does it make sense to replace just the compressor or the entire unit? (note that the former owners appeared to do very little preventative maintenance on the HVAC system.Typically, a contractor will want a whole new compressor and coils but, do some price comparison to find out which will be the cheaper route.
5) If I were to replace the AC unit, would it make sense to replace the furnace as well with a more efficient unit? (note - I do not intend to stay in this home for more than approximately 3 more years so will most likely not reap the financial benefits of the higher efficiency) This is one is rather tricky. As you said, you won't reap the benefits but, many factors come into play here. Will you get a tax break for upgrading to an energy efficient unit? What are the requirements of the tax break (like you have to remain in ownership for X years, etc.. kind of like the particulars involved in the new "gas guzzler - trade in incentives being offered now"? How much value will it add to the resale of your home? My opinion, if the home has A/C, I expect it to work regardless of age, don't care if it's new or old but, it must work. These factors are hard to put a fincancial value on.
As I re-read my reply, I realize that I haven't provided you with much of an answer but, things to consider and research on your own. Sorry I couldn't be of more help.