The first gate checks that Sunrise or Sunset is done, the second checks that Full Moon or New Moon are done. Then you just need 42 draconic to push both open.
Wheel puzzles requires both wheels to be correct to unlock something. There are no other puzzles in the temple, the rest is set dressing. The Tides story goes more into what the temple is all about, and in some cases the tides story is required before you can solve the temple wheels.
The answer for the wheels are "Fii" and "Caputo".