The correct spelling is 'hyphen'.
Yes, it is correct to hyphenate "modern-day" when it is used as an adjective to describe something that pertains to the present or current time period. For example, "modern-day technology" or "modern-day issues."
You don't. The way you have it written in the question is correct.
It's correct but I would hyphenate the two words.
That is correct, but you would hyphenate it as heart-wrenching (causing sorrow or despair).
No, in general it is not correct to hyphenate certain prefixes, co-, pre-, mid-, de-, non-, anti-, etc. You therefore write preordain instead of pre-ordain, antidote instead of anti-dote, and midnight instead of mid-night.
What is a multipurpose system
You do not need to hyphenate.
You do not hyphenate the number.
Don't hyphenate; ongoing is one word.
You hyphenate it only at the hyphen.
I think it's more accepted to hyphenate it.