When we buy say 3/8" hex stock, is 3/8" the diameter of the in circle or the circumcircle?
3/8 is the distance from flat to flat of the hex
-If we buy 3/8" hex stock, I'm assuming we would use a 3/8" hex broach?
yes.
Would we drill a 3/8" hole, then broach it, or drill some smaller hole?
depending on what broach you get there will be a specified pilot hole most will be the broach size rating, but check the broach you plan to buy. mcmaster part# 2875A19 does use a 3/8 pilot
We only have a manual lathe. If we wanted to turn down the ends of the hex shaft so that they fit into bearings, is the interrupted cut hard to deal with
interrupted cuts aren't that bad, as long as you use a good cutter and the correct speed. The trick is just go slow, making small passes. This also depends on your material, if you are doing alum or steal.