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It's a swear-word.
"Падла ты, война!" = "War, you are b*tch"

the earth was falling from the sky (I wonder if it's just allegorical or they meant the pieces soil)

Wouldn't "forward and forward" be better here or is it the same?
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The difference is not very significant, however neither are very natural in English. There is also 'forwards' as well as...

The infinitive is "переть" (there's a touch of rudness in this word), it means to go (move) usually obstinately or blindly, often through some obstacles (snow, rain, for example); in my dictionary...

"полка" is genitive from "полк" which means "regiment". So it means "guys of our regiment".

Верста(pl. вёрсты) is a length unit (= 3500 ft.).