[Rusty]

poner la lavadora = do the laundry
The first sentence is translated "I'm doing the laundry."

The second sentence is translated "I don't like it at all," but that isn't the only thing this sentence can mean (as you stated).

(I've heard a lot of native speakers use «¡No me gusta para nada!» to mean "I don't like it at all!" But grammarians will tell you that's incorrect. In Spain, you'll hear it said, «¡No me gusta en absoluto!», which is an adverb.)

The really important thing you might have missed is that nada can be pronoun (meaning 'nothing' or 'anything' in translation) or an adverb. (It can also be a noun, if preceded by an article.)

If you don't like anything, it's correct to say «No me gusta nada.» Here, nada is a pronoun.

When the lady said her sentence, however, she was using nada as an adverb, describing how much she doesn't like doing the laundry.
If she thought for one moment that whomever she was talking to wouldn't know what she meant, she could have made the sentence more clear by saying «No me gusta nada poner la lavadora.»


«No me lo gusta» is an improper way to say "I don't like it."
gustar(le) algo a alguien
The subject of the phrase correlates to 'algo'. The subject can precede the verb or it can follow it, as in the model above. When the subject is 'it', it isn't said in Spanish (and 'lo' can never be used as a subject pronoun).

The proper way to say "I don't like it" is «No me gusta.»
If there's no referent (your listener doesn't know what you're referring to), you can provide the subject. If the subject is an idea, or an action, state the idea or action using an infinitive, and the subject's object can be a suffixed pronoun (if it has already been established). If it's not already established or understood, it needs to be stated.
«No me gusta poner la lavadora.» = «No me gusta ponerla.»