But aren't the percieved colors redshifted? Aren't these "false color" images just blueshifting the colors back to a visible light image, which is how they would look if they were moving at the same relative speed as us?
I’ve been unable to find any technical reports on the technique, but the gist of it is here: https://www.axios.com/2022/07/17/james-webb-space-telescope-...
No. There are basically two things influencing the colors of galaxies, their age (young ones have bright, blue stars, old ones are more red) and their distance. In the first "unveil" image of the gravitational lens, you can find several extremely red objects (that are completely absent in the HST images if you do a side by side comparison). These are likely extremely distant galaxies where the "blue"/"green" bands in the JWST image (and all the HST bands) are very faint because it falls in the extreme UV in the galaxy's rest wavelengths.
Think of it this way: If you take a picture from a mountain, things in the distance will look bluer than they are when you are close, but does this mean that all the pictures taken from mountaintops have "fake" colours that need to be corrected?
In this sense redshift is just another thing that makes things change colour as the distance increases.
Most stellar objects emit a whole bunch of wavelengths of light, some emit more IR than visible. Interstellar dust doesn't red shift light due to velocity but because the molecules absorb higher energy photons and re-emit lower energy photons.