As far as the US goes, that's incorrect. The issue is a 1A issue, not a 5A issue.
No, it's a Fifth Amendment issue. The Wikipedia article I linked to discusses it. Being compelled to provide a password runs into some of the same problems that compelling self-incriminating testimony does.
search
You're confusing the Fourth Amendment -- which deals with searches -- and the Fifth Amendment. You're right that it's not an issue of protection against illegal searches, which is what one might assume to be the case, but not correct as to the actual rationale that it runs into.