The sentence at hand:
雪がたくさん降っていて、…
My interpretation would be that 「降っていて」 is the て-form of 「降っている」, which is the ている-form of 「降る」.
Intuitively I would understand this as something like "It keeps on snowing", but I'm unsure what the progressive tense of the ている-form means.
I've found a few questions answers regarding the use of 〜ていて in different contexts and different words, but none really seem to apply here.
This is nothing more mysterious than "A lot of snow is falling and ...".
Your grammar analysis is correct that 降っていて is the て-form of 降っている, but there is nothing progressive about this second て. There is no hidden いる on the end. This is just the normal て that joins clauses. There is however a hidden clause that comes after the whole thing. This may be hinted at by the surrounding context. The Japanese like to leave sentences unfinished and let you fill in the rest for yourself; this is quite normal. Without the rest of the context you will have to guess for yourself what might have been implied to come after. For example, if the conversation had been about going shopping it might have been "...and we can't get the car out".
