I was wondering if it is safe to put snow in water filtration and then just wait until it melts down if the temperature is right. Of course, carefully picked, clean looking snow.
Yes. There's nothing wrong with melting snow and then purifying it with a standard water filter. Most of the water in mountain streams was snow at some point anyways.
That being said, this is generally going to be a very inefficient way to make water, and if the temperature is below (or really anywhere near) freezing, you're going to be thirsty. I would strongly recommend carrying a stove instead. Bringing water to a rolling boil kills almost all pathogens. It doesn't remove particulate matter, but clean snow generally won't have many particulates anyways.
A small stove like the MSR Pocket Rocket I have weights much less than a good water filter. Not only does it allow you to melt snow, it also allows you to get a warm drink if you need it, and you can purify regular water by boiling it too.