1. "They set them up specifically for DAA or DAD tuning."

Aha! You are the first person who has said, this, Sean. I just tried tuning an A string to D, and it seemed to be okay. Trouble is, I'm sure I've tried this before and it hasn't helped much. (Strings are brand new today.)

2. Mine was factory-made, not from a kit.

3. Question - if I've bought strings that are slightly heavier guage than planned for, would they 'sit too high' in the slots and be out of tune?

I'm asking various questions as a way of exploring and learning here, as well as trying to solve this particular problem.