The case with an open input in DCV mode is not a good testpoint. The voltage is rather random from a very high input impedance and a tiny bias currents that both have only limits and can change with temperature. +-11.9 V may also be at about the limit what the ADC can still convert (it only gets a +-12.8 V reference and a few percent are lost to fixed parts in the pattern).
It's not random. There's an issue with a specific sequence that causes ~-11.9VDC to appear in the circuit as described,
and be displayed on the front panel. This is not normal behavior, and is clearly the result of
a fault.
U139 (LT1124) is running hot by design. It is a rather fast amplifier with a high power consumption. I consider it a poor choice, but the alternatives with the odd pinout are limited. The LT1013 is also not ideal (more white noise, though less 1/f noise).
A point to check with U139 is whether it oscillates - as a fast amplifier it is somewhat prone to oscillation.
Yes, but we know from other users that mine is hotter than average. 60 to 70C is more common. Mine runs at ~92C after replacement, and was at 102C before replacement. The one I replaced it with is spec'd to 125C, so I'm not worried about the part itself. However, it seems like a potential clue if a part is running 20C above average.
Keep in mind that the partial schematic is not free of errors. The area around U176 - Q184 - U174 is wrong. So it would be more about updating / checking the schematics than actually replacing U176.
I will check them again today when I look for the voltages at U177 and Q156 to see if that leads to more clues regarding the -11.9V source.
3/4 of U176 set the gain of the amplifier and the 4th switch seems to connect one more input.
The self test does not list an error for DCV, so this part should be largely working - at least as much as the selftest can easy check.
Largely working is not necessarily fully functional.

The switching part that could have issues is the JFET base part at the input. This part may very well follow the K2000 schematics, but we don't know for sure. I would especially suspect the mechanical front rear switch and relays.
Yes, I assume you're referring to Q104, Q105, Q108, and Q113? That's on my list to check stuff later on if I don't find the fault in the section mentioned above. I'm working on the assumption that the faults might be related, and that is the part in the block diagram closest to the INPUT that would affect both ACV and DCV. X1 Buffer and BUFCOM might be worth checking out too, but it seems prudent to review the A/D MUX & Gain section first to at least rule it out.
This is from the K2010 service manual:

Thanks,
Josh