"What did you do the day before yesterday?"
Duo's answer is interesting here. Its given you the answer in casual Japanese, incorrectly.
The (no) particle is used as a question marker in casual form speech, you cant really mix the two forms together until you are creating very long and complicated sentences with multiple verbs.
Where duo is wrong here is the conjugation of "masu" form in casual speech. Usually if you were speaking politely (masu form) you would take "to do (su ru)" and change it to (shi) then add (masu). And yes, (ka) is the question marker in polite form Japanese.
So "Will you do it?" would be "(shi masu ka)"
However in casual form Japanese, you keep it as (su ru) and don't add (masu.)
Adding (ta) simply makes it past tense and then adding (no) makes it a question in casual speech.
So it would be "Did you do it?" or "(suru ta no?)" or in this questions case:
"What did you do the day before yesterday?" would be "(ototoi wa nani wo suru ta no?)"
The plain form of する is した、never するた. So in the example above it should be 一昨日は何をしたの？
It's not as simple as just adding た to the dictionary form of a verb, it has to be conjugated. Check out this website for more info on the plain form in past tense: https://www.learn-japanese-adventure.com/japanese-past-tense.html
The の you could call a particle, it's not really a particle. Basically you put this after verbs in questions because the の identifies that the person asking the question is looking for an explanation or reason. Like 何しているの？or "what are you doing?" The person asking is searching for an "explanation or reason" behind what they are doing. You can also change this to 何してんの？for speaking efficiency.
you really shouldn't be using google translate as a guide for Japanese learning, but if you use the kanji you might get a better translation。おととい is pretty common as well.
Anyways, try using jisho instead.
は is a topic marker, and while the topic and subject can be the same thing, they aren't always. The topic is just what the overall conversation is about, person place or thing, subject, object or location that is known information by both the speaker and listener. When you see it you can think of "On the topic of..." or "Speaking of..."
おととい is not the subject here, it is just the context given before giving the actual question so it takes は "As for the day before yesterday, what did you do?"
が is a bit closer to the subject marker. It marks the do-er or be-er in a sentence. It also marks new important information.
If the subject is unknown and needs to be introduced it takes が, if it can be inferred through context of the conversation or was previously mentioned it may become a topic and take は
を marks the direct object of the sentence, the thing that is being acted on.
You know which one to use, not based on what the word it is attached to means but for the role it plays in the sentence. An object will take an object marker, a location will take a location marker, time a time marker, etc.
する・します is a transitive verb "to do", meaning it takes a direct object, a thing that it is acting on. The thing that you 'do'. The thing that you do would be marked with を, and the thing doing the action would be marked with が or は depending on context.
何 here is the unknown thing standing in for the direct object of the sentence in "What did (you) do?" - "You" are the thing acting (and implied through context), and "What" is the thing receiving the action, so を is needed.
何がしましたか "What did (it)?" now marks 何 as the thing doing the action and the object that was acted on is implied through context.