Maybe this is intentional, but the up/down controls felt a bit counterintuitive to me. Up makes the plane go down, and down makes it go up - is that by design?

When you're in a cockpit you're pushing the stick forward to dive, up key on the keyboard is often what we use to represent that but it's not a perfect analogy.

That's how a real plane's control works, so many games do the same.

Indeed. Trim wheels when rolled upward trim for higher speed, i.e. nose down.

Wait, this agrees with GP's experience but not mine. Did OP invert it?