For any given alphabet A, and for any positive integer n, the set of strings of length n over A is a finite set, with (number of characters in A)^n elements.

The set of all strings, of any length over A, is an infinite set, because it is the union of all sets of strings of length n for each positive integer n.

So if you don't know the length of the password, there are infinite possibilities. If you do know the length of the password, there are only finite possibilities.

Which would in turn imply that there is an infinite amount of information in knowing the length of a password - the complement of the set of n-length strings over A in the set of strings over A contains an infinite number of elements, which you can safely exclude now that you know the password is part of the finite set of n-length strings over A.

Only if the password is infinitely long. Which it isn't. The only way knowing the length shaves off a significant amount of time during bruteforcing is if the password is already so short that the time save isn't relevant in the first place.

Absolute nonsense. Apart from the fact that password length is necessarily finite due to memory and time constraints, passwords aren't stored as clear text. You will get hash collisions, because the number of unique hashes is very much finite.

Your argument therefore doesn't apply in this context.