According to Toller :
I removed one wall in preparation to building two to replace it. The bottom
plate was not fastened in any way to the concrete floor. That was nice
because I didn't have to rip it up, but seems inadequate.

Is that considered an acceptable method, or should I fasten it? Is
construction adhesive adequate (should be better than nothing...) or do I
have to go with anchors of some sort; specifically?

I second the notion of using tapcons. Use the larger ones (hex heads). I
think they're #10 or thereabouts. They're easy to install. Bore
some holes in the floor plate (before you install it), install temporarily
and use the proper size carbide bit to drill thru the holes to mark the
concrete. Remove the floor plate and finish drilling. Reinstall floor
plate and insert fasteners.

You can bypass the remove/reinstall step, if you can manage to clear the holes
properly thru the floor plate.

As long as I have your attention. One wall is going to butt up to an
existing wall that is drywalled. I was just going to leave the drywall, and
glue the end stud to it, since it doesn't happen to hit at a stud. Not very
strong, but I don't see that it needs any more than that. Is that
reasonable?

No. You'll be unhappy with the result. If nothing else, you'll have
breaks in the drywall job at the joint.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.