"It is snowing."
So if this translates as snow is falling, do you use the same construction for rain, hail, etc.?
Yes. Pada (śnieg - snow, deszcz - rain, grad - hail). Sometimes just bare Pada is enough. Depending on the season it means rain or snow, but mostly rain.
Jest ciepło/zimno/upalnie/letnio - It is warm/cold/hot/lukewarm
Grzmi - It thunders. We focus on sound.
Błyska (się) - It thunders. We focus on visual aspect.
(Wiatr) wieje - The wind is blowing
That's also correct. You would put more emphasis on the snow than on the falling with that sentence.