Dumb question, pretty sure I know the answer. Which end of the TT & DT goes to the HT? I presume the thicker / painted ends should be by the HT, right?

The painted end is not a reference as different mfg's can use paint to mean different things. Assuming that after the tube is mitered there is a "long" and "short butt, the long one usually goes to the head tube. Most round tubes have butts of equal wall thickness at both ends- when you say "thicker" end, do you mean longer butt, or an actually thicker wall? If you mean thicker wall make sure you haven't just turned a seat tube into a own tube.

FWIW...

From Nova Cycles Supply: https://www.cycle-frames.com/bicycle-...g/TUBES-STEEL/

"The painted end of a tube is the SHORT BUTT SIDE. Make your minimum cut from this side and your adjusting cut on the opposite side."

And from Henry James: https://www.henryjames.com/butt.html

"Tubes with unequal double butting are marked with paint at one end. An asterisk (*) in the chart shows where the tube is marked.

Double asterisks (**) show where external butting occurs."

Balancing butt lengths is also a consideration to throw out there.
Some overall tube length/butt length to design tube length combinations just flat don't work very well.

Depends on the brand. I mark the butts on my tubes so that I don't make mistakes and I can also choose where to cut and how much.