Can I put any word between every and day in these circumstances:
I feel disappointed every exam day.
He brings his umbrella along every cloudy day.
Yes you can.
Every exam day
every day there is an exam
Every cloudy day
every overcast day
Every rainy day
every day when it rains
Every other day
every second day
These all act to describe the frequency of the days you want to mention.
Yes. It acts as a restriction on the phrase "every day", meaning only those days to which that adjective or attributive noun applies.
Except if it's an expletive. Then it's just for emphasis. E.g. "every damn day"