Learn Russian: Получив рукопись из издательства, мы должны выделить ошибки. - Having received the manuscript from the publishing house, we must highlight the errors.

The phrase получив рукопись из издательства is a verbal adverb phrase (деепричастный оборот). In Russian, these phrases act as secondary actions and must always be isolated by commas, regardless of whether they come before or after the main clause.

Why do we use the perfective verbal adverb получив here instead of the imperfective получая?

Получив (having received) shows that the action was completely finished before the main action started. If we used the imperfective получая, it would mean "while receiving," which doesn't make logical sense since you can't highlight errors while the manuscript is still in transit.

Could we use нам нужно instead of мы должны in the main clause?

You shouldn't. A strict rule of Russian grammar is that the hidden subject of the verbal adverb must be the same as the nominative subject of the main clause. Мы is in the nominative case and clearly performs both actions. If you used the impersonal нам нужно, the pronoun would be in the dative case (нам), which traditionally makes adding a verbal adverb like получив a grammatical error.

Why is the word рукопись not in the accusative case, since it's the direct object of receiving?

It actually is in the accusative case! Рукопись is a feminine third-declension noun (ending in a soft sign). For these nouns, the nominative and accusative forms are identical, so the spelling doesn't change here.