A few episodes earlier, in
"Patterns of Force", Spock claimed the odds of another planet developing a culture like Nazi Germany were "virtually impossible." But here he has no problem accepting a culture identical to Earth's ancient Roman culture, right down to the same names of the deities.
Rome did have a sun god named Mithra. Presumably Spock and the others didn't know that although it seems a bit odd Spock wouldn't remember it.
Why doesn't Scotty lock on to the landing party and beam them up? Once Kirk calls him he's got a pretty good idea of where they are, and it shouldn't be hard to locate Spock in any case on the sensors.
For the first but not the last time, Kirk doesn't have the landing party injected with subcutaneous transponders like they did a few episodes earlier in
"Patterns of Force". This would seem to be a necessity on any covert mission where they may not be able to use their communicators, and when even if they die the Enterprise might have to recover their bodies (particularly Spock's!) to avoid a Prime Directive violation.