Is "would" correctly used in the following sentence?

No. You need the subjunctive mood, not the conditional, which would appear in the next clause:

"If the world were to end tomorrow, would the atoms of life from this planet end up on another?"

You could also phrase the sentence, "If the world ended tomorrow...."

The normal phrase is "if the word ended tomorrow" which you can verify by googling it. This is kind of tricky because obviously you don't expect the world to end tomorrow. If it was something possible, you would say "ends." "If the storm ends tomorrow, we will have a picnic." Anyway, using "would" in the if-clause is always wrong, although some people do it.

Yeah, it's fine. Is a good lead-in to a question like, "If the world would end tomorrow, what would you want to do before then if you knew?"

Alternatively you could phrase it as "if the world ended tomorrow" just as easily.

Wenn die Welt morgen zu Ende !

Not really. "Were to" is more common in English.