The sky is not completely transparent. That’s why stars twinkle, because air is moving around. But it’s mostly transparent. How high up do you need to see the stars? On the ground.
As you can see, only certain wavelengths can get through. This assumes no clouds or other junk in the air. The higher you are, the less the available junk in the air. This is why the best ground-based observatories are at high elevation locations. To do better than that, there are airborne observatories: https://www.sofia.usra.edu/ To do better than that, there are space observatories: http://hubblesite.org/the_telescope/
“planets would twinkle due to atmospheric disturbance just like stars, but they appear much brighter and wider than stars, so we don’t see any twinkling.”
Although it’s a satellite and not a planet, I guess the moon’s dearth of twinkle is an easily observable example of that.
@Brian1946 Basically apparent magnitude and angular size is what is important. The bigger and brighter something is, the less likely the object will twinkle.