-Can I put some narrower tires on the 26" MTB wheels?
You can get a pair of 1.5" slicks that work beautifully on a mountain bike. That's what I have used for a couple years now, and just having them be slicks makes a huge difference in the riding experience.

-I am not totally sold on staying with MTB bars because I find that they aren't that comfy on the wrists. Drop? Moustache? What do you think?
I converted mine to drops. This ended up being pretty expensive, requiring a few weird parts (the people at my LBS called it "rollamajigs") and a lot more replacements than I expected (shifters, brakes, cables), but I found it does make my commuter mtb *much* more comfortable.

I think, though, you could just BUY a commuter bike outfitted already for the cost of having the LBS replace your handlebars.

-What's the best way to carry stuff like a change of clothes, a 2 liter of Diet Coke, or the like? I won't need a trailer or anything, but I don't want to run out of room, and I don't want to spend an arm & a leg (as much as I loooooove all the classy canvas/leather touring stuff).
So far I've found my Timbuk2 bag carries tons of clothes and stuff well, but I'd recommend slapping a rack and panniers on that sucker if you want to carry groceries. I can't imagine a 2 liter bottle feeling so great in a messenger bag!

Note on the rack: I found that using the rack on my mtb made it feel very draggy, like I was pulling a dead elephant behind me or I had a flat all the time.