"Snow sticking" describes a very specific phenomenon. When the temperature is well below the freezing point, falling snow is powdery in consistency and it almost seems dry. But when the temperature is closer to the freezing point, sometimes the snow melts and refreezes over and over as it falls. This causes the snow to form heavy clumps, and we then say that it's "sticking" as it gathers on the ground and on surfaces. Rain can't "stick" in exactly the same way, but there are related phenomena called "hail" and "sleet" that can happen when it's raining.

No, you can’t say that. ‘The snow is sticking’ doesn’t sound quite right either, but I can understand what it means i e. it isn’t melting This would not apply to rain, sun or clouds. In these cases, as well as to say that the snow is still falling, you can say ‘it is still snowing, raining / sunny / cloudy’, etc.

I've never heard anyone say that in my life. If snow starts to accumulate on the ground we say it's "lying" or "settling".

Only the snow sticks. The answers you get on this question will vary from country to country depending on what sort of snowfall they experience. I have lived in temperate zones as well as semi-tropical. Here on the southern Georgia coast people have never seen snow sticking so the concept doesn't exist. In the north it simply means the snow remains on the ground - it does not melt. It has nothing to do with "heavy clumps" but I will not argue with the the English. They have their snow and we have ours.

As a Canuk, I would like to add a comment. I live in an area of Canada that us inundated with snow every winter. When snow falls, and the temperature is slightly above freezing, the edges will melt and the snow sticks in larger clumps of flakes and it is a different consistency on the ground. We call it "wet snow" sometimes. It's totally different from powder snow. It's miserable to shovel, but it makes the best snowmen.