Coarse threads are fine, and I believe strong enough. Wait until more professional people answer. They may think otherwise. I believe mine is 5/16-18 or there abouts 8mm.
3/8 seems large, what size is the chuck again. I think this is a 10" or 9" lathe.
yes to the pilot. And with a keyed chuck in a lightly tightened state, not a Keyless which will tighten more, yes. it's a through hole right, so a gun tap will be fine. sprial flute ... not so keen on since the chips (powder with CI) won't evac as well as curly chips. I'd want the CI chips going thru and dropping out the bottom. But that's my choice. I also would make sure you have the chuck loose enough to slip if the CI has a hard spot.... I always try to power tap , except maybe 4-40 and 6-32 and smaller.
That register on the back of the chuck looks like the right size.