The significance of this quote is Montag's realization of what his job is actually about and what it implies. He became a fireman because he was meant to. However, he is now haunted by this new feeling and does not know anymore what is right and what is wrong.
Time contextTime context considers the authors time and place to where he/she receives and conceives opinion from different perspectives and dialogues.
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Yes, unlike a summary you are describing the *whole* book, movie, etc. You include what would be considered spoilers, and how it ends.
To snow.
Why? "It's beginning to snow." makes the most sense. "It's beginning snowing." doesn't make as much sense.