Translation:Then, I will do the laundry.
Without the added subject (He/Her/I) this would translate as "Then do the laundry" which in english isca request or suggestion however the verb form here in Japanese does not support this being a request so the only thing that makes sense is we are talking about something, in the case (He/She/I) will do the laundry
Likewise, that would be a fine argument for including personal pronouns, if not for one thing: if people who are new to Japanese constantly see sentences with "I/(s)he/we/they" being made explicit (in Japanese), they might assume that's just how all sentences go, just like in English, and end up sounding very strange.
Yes, command sentences (called "imperatives") have their own form. For "to do", in the example situation you gave, that would be して. As in: じゃ(あ)、せんたくして.
Also, note that the meaning of "then" is different between Duo's and your own sentence. それから is 'then' in the literal sense of "after that", which is why a comma is used after "Then, ...". If you're using it as a suggestion like "well,/ in that case", Japanese doesn't have an exact word for it, but uses a vocable (a somewhat untranslatable sound that expresses something particular in a language/culture): じゃ.