I can still use the control box even if I switch over to an alternator, can't I?
Only from a decorative aspect. I mean, you can leave the box mounted on the car, but it won't be serving any functional purpose. The alternator has to have it's own regulator (which is usually inside the alternator housing).
If you do replace the control box, try to get one that matches your generator. Later cars used a slightly larger generator, which is frequently supplied as a replacement for the early one. Lucas just warned against mixing them, but I believe that if you use the early box with the later generator, you'll just not be taking advantage of the higher output ability. The danger would be using a later control box with the early generator, which can overload and overheat the generator, shortening it's life.
The fuel gauge doesn’t work and I assumed from the wiring diagram out of the manual that it was probably connected in some mysterious fashion to the wiper ills. I am looking forward to seeing your diagram when the service where it is located is back on line (they are doing maintenance right as I type).
Pretty much the only connection is that they get power from the same fuse. Could be the fuse is blown, or there is a bad connection at the fuse block. There is also a funny little jumper between the two A4 terminals that can sometimes get broken or lost (causing the side A4 terminal to not get power).
I took the wiper apart and whoever was there before me saw it was very worn and didn't bother re-assembling it with the supports for the brushes. As it stands, I think I'll try to find a replacement because this one is, as I say, very worn. There is a shop in UK that sells expensive re-built motors.
TRF has them too, but still expensive ($250 plus core). Might be a few dollars cheaper than your UK source (by the time you pay shipping both ways), but maybe not.
I bought an overdrive transmission from John Esposito at Quantum Mechanics in Connecticut but the wiring remains a bit of a question and John has some personal issues going on so he is unavailable. Before I put it in I’d like to devise a switch that mounts on the shifter. If you have any ideas about that I would be most grateful to learn about them.
There is a special shift knob used on later Spitfires and Stags, that takes an OD switch. I'm not certain how easy it would be to adapt to the TR3 shift lever, but that is where I would start.
However, in all honesty, I like the TR3 OD switch better. The switch on the Stag is both harder to sense position and harder to move than the TR3 switch; especially if you are wearing gloves. The switch itself seems kind of fragile, as does the thin plastic knob cap. And, when a buddy of mine did something similar on his TR6 (by having a Spitfire OD shift lever welded to his TR6 lever), he had trouble with the wires shorting inside the shift lever. The TR3 dash switch is functional, rugged, durable, and quirky. Suits the car much better!

And I love how, with practice, it seems to almost read my mind. I don't have to think about it at all.
OD wiring is pretty simple, but it helps if you follow the inset on the diagram I linked to above. The wiring shown on later diagrams (and the Dan Masters diagram) will work, but is more complicated IMO.
I do suggest one modification, though : Add an in-line fuse holder with a 5 or 10 amp fuse in the line labelled "45" in the diagram above. It will help protect not only the wiring to the relay and solenoid, but may save the solenoid itself if it gets jammed somehow and can't pull in all the way. I had a piece of gravel get inside mine once, and it burned up the solenoid.
Feel free to ask any OD wiring questions; but it would probably be better to start a new thread with an appropriate title.