"It is cold, and it is snowing."
Translation:Il fait froid et il neige.
It is cold. Cold is an an adjective. The expression "Il fait" (it makes) requires that is it followed by an adjective. So the expression is "Il fait froide" (it is cold.) Il y a is used to describe what you can sense - a noun, the rain, the snow, the fog. Cold is not a noun so you would not say "Il y a froid"
"Le vent" is a noun. It means "wind". You say "il y a du vent" to mean "there is wind" or "it is windy".
"La neige" is a noun, meaning "snow". "Neiger" is a verb, meaning "to snow". You say "il y a de la neige" to mean "there is snow". You say "il neige" to mean "it is snowing."