"I do not wash my clothes."
Whether it is positive or negative （洗います/洗いません）, 服 is the direct object. So gramatically the particle を is needed to describe the role of 服: 服を洗います or 服を洗いません.
は is to bring up the topic, as already discussed above, and it is added after a phrase (unlike を which is after a noun equivalent). In this case, the は is added after 服を becoming 服をは. In modern Japanese, は supesedes を, so it becomes 服は. So 服は is 服を+the indicator to mark 服を as the topic.
は can supesede also が, so 服は can be either from 服を or 服が depending on situation.
that make sense, I will edit my comment out so people don't get the wrong idea about how this particle works. Thanks for the explanation, it really cleared out a lot of doubt I had.
I originally got my explanation from a discord group, but it was over simplified, so I might have gotten the wrong idea.
As I understand it after crossing multiple sources (I am not a native speaker nor a teacher):
「は」is a topic marker, the closest English translation is "as for"
「って」is another famous topic marker, closer to "speaking of" only used in spoken language.
Now, any of the four usual particles 「が・を・に・で」, can be transformed to a topic marker, but only 「が」 and 「を」 can disappear completely:
Once a topic is established, it is implied for the next following sentences.
Wa - topic marker, "theme" or "as for..." Ga - subject marker, SOME times inerchangable with wa but implying something a bit different Wo - object marker, used when targetting a specific noun with a verb or an action
Both of aforementioned sencences are fine but the first one is much more casual... a lot is inferred in everyday Japanese and often some particles or topic are dropped cause it's understood anyways. It's usually okay to drop watashi as well (unless you wanna emphasize it) cause people would likely understand that you are talking about yourself. The second sentence is perfecltly correct gramatically but there's no need to be so specific when it comes to casual speech ... The sentence is randomly generated anyways, it's just for the purpose of the lesson. But I think it would be good if Duo specified if they require polite or casual speech from you and so on... :)
You got these topic and particle right however I think your verb tense is off. -Masu and -masen are considered non-past and can talk about present or future actions but not progressive. So in this case the sentence 服は洗います Would either be: "I wash the clothes." or "I will wash my clothes' Progressive (washing) wouod require the -te/て form + iru/いる
"Don't wash clothes" is in imperative form so it would be 服を洗うな. "The clothes are not washed" is in passive form so it would be 服が洗われません. You won't know whether it is "I" or "He" from the Japanese sentence. The subject is omitted and this is how Japanese language works. You assume the subject is "I" for declarative sentences and "you" for questions or requests etc. using common sense.
While the subject is usually omitted in Japanese, an English statement needs a subject.
Omitting the subject in English changes the meaning of the sentence from a simple statement to a command. (Telling someone else to not wash). This would be 服を洗わないで・洗うな in Japanese.
Without any provided context, Duo assumes a two-person conversation, where statements default to the speaker "I" and questions to the target listener "you".
"I do not wash clothes" "Do you wash clothes?"
"The clothes are not washed" would be a different structure as well, with the clothes being the subject of the passive form 服は洗われません