MUR460 diodes get extremely hot - what am I missing?
First of all, paralleling diode doesn't work very well. Unless the Vf are matched current will prefer one of the two. It sort of work, but not expect a 50% load sharing.
Fast diodes will work perfectly in DC, simply they are not optimized. Maybe they only have slightly more Vf (and are more expensive). Remember that dissipation in diodes is Vf times I. Schottky diodes also have substantial leakage which often predominates when you use them as switching diodes, but this is not the case.
110W at 18V (assuming the worst case) is slightly more than 6A; a cautious derating would choose an 8A silicon or Schottky rectifier for this application: this is a power oring condition (and I would strongly recommend a MOSFET and an oring controller for this amount of current) so let do some math on some plausible diode.
As a Schottky a choice would be for example an SBR1045, a chunky axial diode. Average I=6A, about Vf=0.4V at the operating point, about 25°C/W to ambient. That would dissipate 2.4W with a temperature rise of about 60°C. It could work, if you don't touch an 80°C diode body. Tj max in DC is 200°C so don't worry for the diode.
Now for the silicon choice: MUR820, the obvious stepup from your choice: the TO220 offers a 75°C/W free air dissipation, Vf at about 0.85V (at cold). It would dissipate 5.1W so there's no chance it would survive without heatsink (like most TO220 at their intended power).
In an oring controller situation, the main power loss would be the RDSon of the switching MOSFET. I'd take as a comparison a TO220 too but this would be in practice done with a DPAK if not with a smaller SOT package. But I'm lazy and I don't want to keep half a day choosing a MOSFET. Take the IRFZ24, I have a bag of these. Survives with 175°C at the junction. Nominal 70milliohm, about 2.5W of dissipation at 6A. Still not completely safe without heat sinking and in fact slightly worse than a Schottky. Well, it actually cost less than that diode, and it's not even a low gate drive MOSFET.
However these day steady state optimized MOSFET can go down to about 1 milliohm, so you'd be dissipating only a quarter of Watt.