Kip's answer is similar to the one we often see. Michael Swan, for example, (Practical English Usage, 2005.611) writes:
We prefer which when we have a limited number of choices in mind.
We've got white or brown bread. Which will you have?
(More natural than ... What will you have?)
Which size do you want - small medium or large?
When we are not thinking of a limited number of choices, what is
What language do they speak in Greenland?
(More natural than Which language ...)
What's your phone number? (NOT (!)Which is your phone number?)
That seems moderately clear, but note that Swan uses the word preferred, not mandatory.
Huddelston & Pullum (The Cambridge Grammar of the English Language, 2002.903-4) note a similar difference in usage to that noted by Swan, but also say that in some cases, "The difference between what and which is effectively neutralised".
Now, let's look at your original questions:
Which/what day is Friday? — It's the fifteenth.
Which/what day is the first of November? Is it Saturday?
Which/what month is the warmest in your city? — It's July.
Which/what season is it now in your city? — It's summer.
As Huddleston & Pullum say of different sentences, "... which encodes the fact that the choice is from an identifiable set" [thirty-one dates, seven days, etc] "while what doesn't. but as that is part of background knowledge it doesn't matter from a pragmatic point of view whether it is encoded or not".
My own somewhat cynical view is that grammarians have tried to make a usage difference which many native speakers do not feel. Some native speakers use which if there is a very clear limit to the number of choices, and what if there is apparently no limit. Others use whichever/whatever wod comes to mind first. Very few people will be upset if you make the 'wrong choice. Most won't even notice.
In your sentences, I'd use what in the first two (I think) and either word in the third and fourth. Others might well make a different choice. Sadly, I don't think the bonus is going to attract a definitive answer.