Yes, it's possible to use it as a verb.

I agree with 'sleeting,' too. I think I hear the word most often (or heard, especially when I was younger) as an odd adjective: "-It's snowing again. - looks more sleety." "It's been sleety out there all afternoon."

We do, all the time: it is raining, it is sleeting, it is snowing... Perfectly correct.

No I don't. People just say it's snowing. The only time I hear the word sleet use is from the weather reporters. example They say "Tonight we have a chance of rain mixed with sleet."