"What day of the month is it tomorrow?"
Agree. You do not know if they want you to ask with all the English words or ask in a shorter version, like 'what is tomorrow's date' (which seems to be what the characters are saying).
No, the characters here only ask about the day part. For "date", it should be ji3 yuè ji3 haò.
I believe you could use '明天是什么日子‘。 Literally 'tomorrow to be what day'. I think 明天几号 is a colloquialism.
With "明天是什么日子" you're asking if it is a day of significance tomorrow, but not asking for the date.
We do not speak of the day of the month in English - when we mean: What is the date?
This question was not formulated by a native English speaker.
You could use this type of language when referring to the female cycle or Christmas or payday, but even in that context it is poor English.