As I mentioned in a comment that may or may not get deleted, I think the question itself is very broad. There's just no way to reasonably prove that some particular color isn't possible. That said, we can show that most colors are possible, as long as you don't need it to be some really specific shade.
Red from dust, yellow from dust, green from storms, and blue from Rayleigh scattering are all perfectly normal here on Earth under conditions that could conceivably be permanent on another planet. Purple is probably possible with a slightly hotter sun and some atmospheric dust that absorbs yellow and green light.
The reason our sky is blue is because of Rayleigh scattering, which preferentially scatters shorter-wavelength (higher frequency, higher energy, blue-er) light. That will be true of any atmosphere, regardless of composition, but only for the colors that actually survive the trip.
The first part of having colors survive the trip, is they need to exist at the beginning. A star's temperature determines what colors it outputs. See black body radiation topics for more information. A "cold" star (around 3000 K or 5000 °F) will be red. Meaning your sky (and everything else you look at) will be red. As the star gets hotter, the color will progress through orange and yellow, then start becoming whiter and whiter, but the sky will always tend towards the shortest wavelengths visible, so it could look red, yellow, green, or blue. If you make the sun hotter than ours, you could conceivably get a purplish sky, although oxygen tends to absorb light in that part of the spectrum (and we can't see a huge range of purple colors -- most purple colors are the absence of green).
Note that for most of these sky colors, the planet won't be incredibly nice visually, because without artificial light sources, you'll get monochrome or duochrome everything. So if you want normal-ish world lighting with a different-colored sky, you have to rely on chemicals in the atmosphere.
There are two methods of changing the atmospheric color through chemicals. One is by changing the gases in the atmosphere itself (fluorine turns the atmosphere yellow, while chlorine turns it green). The other is by constantly kicking up a bunch of dust of a specific color (red dust will absorb the blue and green light, so only red light is able to refract down to observers on the ground).
Both of the gases listed seem to be toxic in the concentrations needed to change the sky color, so it might be safer using dust particles. Mars is a reddish-orange color because of the oxidized iron in its dust. So adding extra oxygen won't change the color much. Sand on Earth is generally a desaturated yellow color (also known as shades of brown), so anything in that range is clearly possible. Green skies can occur on Earth near thunderstorms (it seems particularly prevalent if there's hail but a quick Google search says we're not really sure what causes it), so a planet with a lot of upper-atmosphere storminess could give green (of note, this is thought to relate to ice in the atmosphere, which could be related to your very cold planet).
Again, the further you are from an Earth-like blue sky, the more of an effect you'll have on anything at the ground level. So bright, pure yellow skies will mean red and green stuff shows up decently, but blue objects will look black. But a faded yellow (which means there are other colors besides yellow, but yellow is the brightest) will be perceived as a yellow sky while still making ground objects relatively normal. Note that even our sky isn't really blue -- it's a faded (whitish) cyan/blue color. But you can still get a reasonably pure sky color without sacrificing ground visuals too much.
Of note, the denser the atmosphere, the more Rayleigh scattering has an effect (and the less relative effect the dust particles have). So for humans to live, we need something more dense than Mars, which means the atmosphere can't be as red, but it can still be fairly red to orange, especially with a mildly colder star. Purple skies can exist with any compound that mostly absorbs green, but I'm not sure what that would be. All the purple compounds I looked up react with oxygen to turn some other color.