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As KenGee said, there are no silly questions.
On this: "But doesn't this also mean that waves in the ultra-violet spectrum would also be redshifted into visible light?" you are 100% correct.
If we call our Sun 'yellow', then as it is redshifted more and more it will appear 'orange', then 'red', then a very dim red, as the peak of its spectrum (around 480 nm) moves further and further to the red. And as more and more of its ultraviolet spectrum comes into view (so to speak). As the Sun a very dim, comparatively speaking, in the ultraviolet, its colour will look more and more red.
There are stars - very hot ones - whose spectral peak is in the ultraviolet; we usually call these 'blue'. As they are redshifted, they simply get brighter; then, as the peak comes into the visual range, they start to look 'white', then 'yellow', etc, just like the Sun (but you need a much higher redshift for these to become 'red').
There are some AGN (active galactic nuclei) whose spectral peak lies so far into the ultraviolet that it's almost into the x-ray part of the spectrum. These objects are often seen as quasars, and sometimes at very high redshift (>~7). They look red, but not because their spectral peak has moved that far, but because all light with wavelengths shorter than 91.2 nm has been absorbed!
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