무슨 요일이에요?

Only with very close friends. It's the 'Banmal' (informal) version of the question.

Simply say the day name + 이에요. For example: '월요일이에요.'

No, '무슨' is a determiner and '요일' is a noun. They are written with a space.

Because '요일' means 'day of the week'. The first syllable (월, 화, 수...) tells you which specific day it is.

You can point to a calendar and say '오늘 요일?' but it's much better to remember '무슨'.

It is '이에요'. '이예요' is a common misspelling even among natives.

No, for holidays like Christmas, ask '오늘 무슨 날이에요?' (What day/occasion is today?).

Korean doesn't typically pluralize nouns like this. Context tells you if it's one day or multiple.

You can say '무슨 요일들에...?' but it's more common to say '어느 요일에...?' or '무슨 요일에...?' and let the context handle the plurality.

Yes, the basic calendar terms are identical in both North and South Korea.