The red trails behave exactly as you say when I turn down the
brightness, I guess that's a bad sign. It seems weird that low red
emission creates more red on the screen, but I don't pretend to
understand how the tube works.

To make it simple, it is not that there is more red. The problem is that
the red that is there can no longer be cut off or focused properly at normal
brightness levels..

I'll fix the bad solder joints and replace those couple of resistors,
and see what happens.

Replacing the resistors might help, but the new ones will likely overheat as
well. The 12K resistors are a series parallel circuit so you would need to
replace all 4 of them. Interesting the total of all 4 is only 12K. I
suppose it was cheaper to use 4 smaller resistors than one higher wattage
resistor. May have also been a heat concern.

There is a small amount of documentation glued to
the inside of the set that has a bit of diagnostic information; I'll try
to read that a bit more closely and see what kind of voltages happen on
the CRT neck board.

The neck board voltages run from under 5.0 volts to over 5 thousand volts.
The voltages on the collectors of the video output transistors are around
100volts. One side of those 12K resistors goes to the 193volt source. The
heavy (usually black or red) wire going to the board is focus voltage
between 5 and 6 thousand volts.

On a side note, I read on the Internet (so I know it's true?) that you
can do things like fiddle with the heater voltage to help the tube along
for a bit longer. Is this really safe? It doesn't strike me as a good
idea, but I like to explore all of the options (at least the good ones).

Yes you could raise the heater voltage and that might help for a while.
But, the heater voltage on your set comes from an internal winding on the
flyback transformer (high voltage). You would have to disconnect it and
replace it with the correct number of windings around the transformer core.
To many turns the voltage would be to high and tube would be instant toast.
I have seen tube brightners for this. They install between the flyback
winding and the tube heaters.
If you had one you could use a CRT rejuvinator on the tube. Neither
solution is going to provide long term reliability. Both could instantly
kill the tube and may require a grey scale setup (adjustment of all the
controls on the neck board and one on the main board) afterwards.

You might want to save your money and put it on a cheap 25" set to last
until you can get that flat screen high def LCD monitor for the wall.

Bill
Christian Technology