It happens with satellites all the time.

On the Huygens probe into Titan, it was realized that they had forgotten to account for the Doppler effect properly, and that the signal from Huygens would not be received. They figured out a spacecraft geometry that reduced the Doppler shift enough to avoid the problem. https://www.thespacereview.com/article/306/1

And of course that's how a radar gun measures how fast your car is going.

But yeah, radio communications with airplanes don't have to worry about it, as it's too small to matter and they can tweak the frequencies if they need to to stay locked on to each other.

You can buy an off-the-shelf Stirling engine two-stage cryocooler giving 10-20K. Radio astronomers use them all the time.

You put in about a kiloWatt of power and get about 2W of cooling, so use a good Dewar. They use Helium as the working gas.