Learn Dutch: Hebben jullie elkaars bureaustoel gepakt? - Did you take each other's office chair?

No, elkaars is a fixed word that always means "each other's". Just like the regular pronoun elkaar (each other), you just add an -s to make it possessive. It never changes form based on the gender or number of the noun that follows it.

Since there are at least two people and two chairs, shouldn't it be bureaustoelen?

In Dutch, when a plural subject acts on something where each individual is dealing with exactly one of that item, we often use the singular form of the noun. Since each person took exactly one chair (the other person's), bureaustoel stays singular. You will see this a lot with items of clothing or body parts, too!

Could I use genomen instead of gepakt for "taken"?

You could, but pakken (to grab, to fetch, to take physically) is much more natural here. Dutch speakers strongly prefer pakken when physically taking an object with their hands, while nemen is more often used for abstract things like taking a decision, taking your time, or taking public transport.

Why is gepakt at the very end of the sentence?

This is the standard word order for a yes/no question in the present perfect tense. The helping verb Hebben goes at the beginning to form the question, and the past participle gepakt gets sent to the very end of the clause, wrapping around the object elkaars bureaustoel.