Translation:Do you stay with me until my father comes, please?
You're welcome! So, as we noticed, "do you stay with me until my father comes please" doesn't really sound right but "stay with me..." would be fine, also "will you stay with me...", "would you stay with me...", "can you stay with me..." or "could you stay with me..." would all also be perfectly good.
This may well not be a thing grammatically but I took it as an imperative question. Yes, I know that doesn't make sense, but it made sense to me thinking of a child making a half-request, half-demand. "Stay with me, please?" (simplified from the version here just for illustration) seems to me to be perfectly natural, even if not the height of grammatical formality.
According to conjugare.ro, ”rămâi” is both the 2nd person singular present form and 2nd person singular imperative form of the verb ”a rămâne”, so it means both ”you stay” and ”(you sing.) stay!”. So the answer ”You stay with me (...)” that Duolingo requires is not strictly wrong if you disregard the ”te rog?” part of the sentence, but it makes sense in its imperative form only if Duolingo changes the best answer to be ”Stay with me (...)”, i.e. imperative form. Check the conjugation here: http://conjugare.ro/romana/conjugarea-verbului-r%C4%83m%C3%A2ne
The sentence is badly pieced together and does not sound like a question. But if you just read it without listening it is correct and it means "Will you stay with me until my father comes, please?"
While "rămâi" and "vine" are in the present tense, it is obvious that they will happen in the future, so we can't say "do you stay..."
A couple of the most natural English translations that seems to encompass this are:
"Please could you stay with me until my father comes?" "Could you please stay with me until my father comes?"
While the please falls in a very different place in the English compared to the Romanian, this seems to capture the whole meaning.
'other' means that there is a second or third choice. So you say the 'other' way of saying it is 'Will you stay with me until my father comes?' Which I would like to be the translation Duo has that is accepted, as with the 'te rog' on the end it doesn't make good English English.
Do you stay with me... might just make sense in English - if the person asking is new to this and asking about the habitual pattern (that the other person has perhaps done with others or which is set out in rules), then adds a plea at the end (te rog). This is the interrogative, I guess (I was never really taught any English grammar associated language to speak of!)