Mostly the 을 referring to the object is omitted in informal speech. (Basically always)
The particles are often left out when the meaning is clear.
In conversation we can drop a lot of the grammatical markers. I am surprised that Duolingo did not include it though.
I think the subject marker is not as necessary as the object/topic marker.
no because you want generality here
어젯? I thought it was 어제? Is this similiar to 거 and 것?
Actually they are different. 어젯밤 means "last night" (literally combining 어제 (yesterday) and 밤 (night)). The ㅅ is often added in Korean when forming compound words.
Should it be 라먼는 or just 라먼 first instead the other way around?