For recent times, you can also compare the dates of the C14 with other methods like counting tree rings, or the date of a total eclipse and check the calibration.
2) You are almost right. The tides are not produced by the gravity of the Moon, but from the differences in the gravity of the Moon in the water that is nearby and the average of the Earth.
You forgot to include the centrifugal force [when you are in the non-inertial frame frame that rotates like the Earth-Moon system https://xkcd.com/123/ ]. The centrifugal force is bigger in the water that is in the more far from the Moon and again the difference creates the other tide.
3) The sky is blue because the single molecules in the air disperse the blue/violet color more than the other colors. There are many ways to produce colors. In this case the light is dispersed by the whole molecule.
A different method to produce blue is using a CD to produce a rainbow and the using a slit block the other colors. Some birds and butterflies use a somewhat similar method. [Not very similar but closer to the CD method than to the air method.]
The blue in the die for cloth uses another method. You make a long chain of conjugate chemical bounds C-C=C-C=C-C=C-C, and pick length and atoms so the electrons absorb the colors you don't like and transform the energy into heat.
I'm probably forgetting a few more method, there are many of them, so it's interesting to understand which of them make the sky blue.
*) These are good questions. My explanations are not 100% complete (and probably not 100% accurate) but I hope you can fix the holes.