Josh thank you very much for all those corrections, Merci bcp!!

"Would you shut the window?" is a way of politely asking someone if he/she would be so kind as to close the window. This sentence is grammatically correct. "Could you shut the window?" is a way of asking someone if he/she is capable of closing the window. In the context you are using, this sentence would be grammatically incorrect. Even native speakers get this wrong the majority of the time.

Would is more polite, but both are used and mean pretty much the same thing in common use. Could technically means "to be able to" or to "have the ability to" in that context, but again...people say that all the time.

Both can be used in a question, but would is the polite way of asking, it is requesting.