Learn French: Qui fait la lessive à la maison ?

When qui (who) is the subject of a question, it takes the third-person singular form of the verb, just like il (he) or elle (she). This is true even if you expect the answer to be more than one person! That is why we use fait here.

Does faire mean "to do" or "to make" in this sentence?

In this phrase, it means "to do". The verb faire is very versatile and covers both "to do" and "to make". French uses faire for many everyday chores, such as faire la lessive (to do the laundry) and faire le lit (to make the bed).

Why doesn't this question use est-ce que or a flipped word order?

You do not need est-ce que when the question word is the subject of the sentence. Just like in English where we say "Who does...", in French you simply put qui followed directly by the verb: Qui fait....

Could I use chez to say "at home" here instead of à la maison?

You could, but the phrasing changes. À la maison literally means "at the house". The word chez means "at the home of" and must be followed by a person. So, you would have to say chez nous (at our place) or chez moi (at my place) instead.