(1) How are handlebars made? Do they start with a piece of metal pipe and bend it to a precise shape, and if so, how do they relieve the stresses introduced by bending? How do they make the clamp part of the handlebar thicker? On old handlebars, it's obviously a sleeve added on to the basic shape, but what about newer bars which just bulge smoothly in the clamp area?

Some bars are heat-treated to releive stresses, most aren't. Bar clamp areas are often hydroformed (think blowing up a balloon) inside a bulged mold. More expensive handlebars are sometimes 'drawn' leaving a thicker section in the higher-stressed middle and thinner elsewhere. Hydroforming tends to do the opposite (think about that when using cheaper thin-wall bars).

(2) Why are spoke nipples made of brass? All other highly-stressed threaded fasteners are steel on a "typical" bike, and almost everything else is aluminum. So why brass for spoke nipples? My best guess is that it has to do with corrosion in some way.

Brass is a low-friction material, plus it tends not to corrode too much.

(3) How do they put cartridge bottom brackets together? I spent a while staring at a Shimano UN72, and for the life of me, I can't see any seams or joints on the shell. On the other hand, on the Nashbar cartridge BBs, the bearing cartridges are clearly press-fit onto the outside of the shell.

Press-fit together by big machines.

(4) Is there any way to repair inner tubes with cuts right next to the valve stem? I have about 10 of these, and would love to know if there is a way to fix them...

No reasonable way to fix them, although I've got a couple of 1950's British cycle mechanic handbooks that show how to remove the entire valve and replace it with one from another tube (not much money around then, I guess). Deburr your rims and change your rim tape to stop that happening.