If I understood correctly, in most situations where hashing occurs, the Bitcoin protocol applies double hashing: sha256(sha256(input)) rather than sha26(input).

Why is this, doesn't this actually reduce entropy?

As far as I know, we cannot guarantee that there aren't many different inputs x and y which have different hashes, but the same double hashes. That is, sha256 is not a one-to-one mapping on the 256 bit space, right?