Can we say" yağmur yağiyor değil"also instead of yağmiyor
No. Değil is used to negate nouns and adjectives. Verbs have their own negative form which involves adding -mez, -maz, -me, -ma or -m- at the end of the stem.
Can this also be "It is not raining, but (in fact, it's) snowing." If not then what would it be?
Yes it would mean exactly what you said. Otherwise it doesn't make sense much anyway. Why do we use comma instead of starting a new sentence? To create the connection (but) that you are talking about.
Thıs really helps! Teşekkür ederim!
wrote the correkt answer and was marked incorrekt with the correkted text identikal to my answer.
kar + yağiyor = rain
Yağmur + yağiyor = snow
So kar = rain
Yağmur = snow
Yağiyor = fall down ?