1) How we know standard atmospheric refraction is a real phenomena and 2) Why you used a value of 7/6*R for it [in the Metabunk curve calculator]

Atmospheric refraction is measurable when observing stars. Stars visually appear to rotate around the Earth once ever sidereal day - and they do this at a constant speed. However as they approach the horizon they seem to slow down a little, basically staying visible longer than they should. When they reach the horizon their actual calculated position is a bit over 0.5° below the horizon - a bit more than the diameter of the sun. This amount matches what we have empirically found about the refractive index of air at various temperatures and pressures, and what we have empirically found about the change in temperature and pressure at various altitudes in the atmosphere, and what we have empirically determined to be the radius of the Earth. The refraction of terrestrial objects differs from stars in that only part of the atmosphere, whereas astronomical observation passes through the entire atmosphere. The value of 7/6*R is just a rough approximation of the effect of refraction on the apparent radius (R) of the Earth. It's very approximate as the atmosphere is quite variable. Basically though you do the same calculations, but the larger radius gives you the reduced curvature. I initially got the 7/6 value from: http://aty.sdsu.edu/explain/atmos_refr/horizon.html The main ways in which observations will differ from the standard refraction are that the temperature near the ground often varies greatly from a few meters above. Also the lapse rate (how quickly the air cools) may not be linear, especially at low altitudes. This can result in vertical stretching and/or compression of the image. If you want more details, including empirical observations, I recommend this set of tests by the USGS: https://www.ngs.noaa.gov/PUBS_LIB/ResultsOfLevelingRefractionTests_by_NGS_TR_NOS92_NGS22.pdf

Here's a great video showing the effects of refraction on stars, the sun, and the moon as they get closer to the horizon. Source: https://vimeo.com/188149183

There's always some refraction yes, because of the pressure gradient of the atmosphere. The actual amount and nature of this can vary - like if there's a temperature inversion. https://en.wikipedia.org/wiki/Horizon#Effect_of_atmospheric_refraction

I made a video analyzing the relative slowdown of stars as they get closer to the horizon: Source: https://www.youtube.com/watch?v=m-xXhrTG3Sk Summed up in this image: What I did was track one star and put a red dot (and green horizontal line) where it was every one second. I then took the first two positions and used them to extrapolate expected positions at a constant speed. The results show that the star is already slowing down a bit, and the slowdown rapid increases as you get closer to the horizon.

Looking for more derivations of the 7/6 (1.1666) value. https://en.wikipedia.org/wiki/Atmospheric_refraction For a standard atmosphere at mean sea level, P= 1013.25, T=288.15°K (15°C), dT/dH = 6.5/1000. Hence, k = 503*1013.25/288.15^2*(0.0343 + 6.5/1000) = 0.25, so Reff = 1.333*R, as opposed to 1.166R Is the seconds equation supposed to use the "smaller" k? Did I make a mistake? Reference 11 is attached, in which Andy Young says: 1/6 = 0.16666, giving Reff of 1/(1-1/6)*R = 1.2*R. The 7/6 is just a ballpark, but it would be good to to verify the derivation of this number (and clarify which "k" is used. It's probably in Dr. Young's document.

Different people use different numbers. I wonder if the calculator should show a range. Maybe even generate a diagram of horizon lines. http://www.aboutcivil.org/curvature-and-refraction.html So they say "14% of the effect of earth curvature.", is this the same as using a radius of 1.14*R? Not really clear what they mean there.

I think they got that wrong for a start. Even if the temperature is constant with height the density will drop due to pressure drop. A typical lapse rate, which is limited to the dry adiabatic rate, would decrease refraction somewhat.

But if you've got a warm region lower down the expansion of the air due to heating reduces the density, cause the bend upwards. Hence some types of mirages.

But the air is unstable, and convection and mixing generally occurs. A hot road is slightly different in that inversion can be stable in a very thin layer, the thermal boundary layer, I think.

Here are two photos of Newquay Cards. One is from 9metres asl from the stone pier at Aber and the other from 51 metres asl and 1 mile more distant but in line. There is a mirage in the lowerelevation photo. The air was about 26C and the sea probably 10C or so cooler.