Sorry for the rapid update, but a few other notes:
This may be a pretty obvious step, but I kind of picked debris out of the melted wax with a toothpick (bits of singed wax or paper, bits of wick, whatever might have been stuck to it). And rather than pouring messily from the bowls I'm using (which are metal, btw), I'm using a little plastic thing I got out of the recycle bin as a pour scoop. It's working okay, not melting, not burning my fingers, etc. Wax tends to have a pretty low melting point, so the temperature it's at hasn't been harmful to anything else, so far (the cardboard tube isn't catching fire, the box top isn't getting melted onto anything, I can move them comfortably, etc.).
I'm trading the bowls while they're still hot from the top of the double boiler with one of those oven gloves, so no steam scalding.
I poured my first little layer of wax (probably not more than a cm thick) while just pressing the tube down against the box top, since it wouldn't stay stuck all the way around from just my wax dipping, and this pouring sealed it down fine--it didn't leak. I just made sure to kind of brace it a little while it cooled a bit. I also trimmed the tube so that it was a little shorter than my wick, and I draped my wicky guy over a pencil that I sat across the center of the mouth of the tube, so it would stay pretty straight upright in the wax.
And finally, if you're interested, here are some of the sites I looked at, which all had different bits of usefulness, and all assumed different levels of preparedness:
Presumably more to come, when candling has ceased! My first one is about as tall as it ought to be, and I think I should probably wait on doing any more until I see how it behaves, but... but.... I have more coated box lid, that might be more like the inside of a milk carton, and that could probably be wrapped into a thicker, pillar-y shape... And.. and I've got all this white and creamy unscented or vanilla scented wax.. and I go through sooo many of those..... And, and, I've got things I could roll a finished candle in, and... . .