Does refresh rate affect a calibration?

That's a definite maybe.

Every manufacturer does different things, I don't know the specifics about your set. The only sure fire way to know is to test.

On windows that are several different ways to change the refresh rate so you should be able to match the signal for either a blu-ray or cable box.

The real issue is how the TV is setup to operate. A 120 Hz or 240 Hz display doesn't necessarily accept 120 Hz or 240 Hz INPUTS. Instead, it accepts 60 Hz inputs and converts those to 120 or 240, sometimes using motion-smoothing processing (sometimes not). These same TVs may also accept 24p inputs and convert those to 120 or 240 internally.

What this means is that if we ever get a 120p video source (or 240p), today's "120 Hz or 240 Hz" TVs will NOT accept those inputs. You MIGHT be able to make a TV that accepts 120p or 240p inputs, but I don't think anybody is making such a home video display today. There might be computer monitors that accept 120 Hz or higher video rates.

Do our eyes see color differently than our equipment/software when refresh rates are changed?

Intensity might be slightly different, but chromaticity would not.

.What you see depends on how the higher refresh rate is implemented and/or what settings you've made to the display. Some high frame rates insert black frames between each "normal" feame. Obviously, that probably won't look as bright initially, but you might be able to compensate by raising contrast or backlight or both. For example, 240 Hz with black frames would be frame1-black-frame1-black-frame2-black-frame2-black, etc. There would be 120 frames per sexond plus 120 black frames per second.

There might be SOME meter out there that has problems with some refresh rate or another, but I've never seen one myself. If the display in the above example measured 30 fL with no black frames inserted, if you switched to a mode where half the frames were black, you'd expect 100% white to drop to 15 fL. If you want 30 fL you'd have to experiment with Contrast and Backlight to get back to 30 fL. Color shouldn't be an issue one way or the other. And I don't recall ever seeing a display that measured differently at a high frame rate vs. 60 Hz, not that I go looking for that, but I do sometimes check things like that the first time I see a display.