Learn French: Nous déballons les lave-linge. Est-ce que tu t'en occupes ? - We are unpacking the washing machines. Are you taking care of it?

Lave-linge is a compound noun made of a verb (lave) and a noun (linge). Under traditional French spelling rules, the verb part of a compound never changes. The noun part only takes an '-s' if it logically makes sense as a plural. Since linge (laundry) is treated as an uncountable mass noun here (the machine washes 'laundry', not 'laundries'), it stays singular. Therefore, les lave-linge takes no 's'.

What exactly does the pronoun en replace in tu t'en occupes?

The pronoun en replaces a phrase introduced by the preposition de. The underlying verb is s'occuper de (to take care of). Instead of repeating what needs taking care of (e.g., t'occuper du déballage or t'occuper des lave-linge), you use en to mean 'of it' or 'about it'.

Could I use ouvrons instead of déballons to mean 'unpacking'?

You could say you are opening (ouvrir) the boxes, but déballer is much more precise here. Derived from balle (a bundle or package), déballer specifically means 'to unpack' or 'to take out of the packaging', which is exactly the action you perform with brand new appliances.