With respect to the exclusionary rule, does the Constitution specifically include it in the Fourth Amendment?
No, the exclusionary rule is not specifically included in the Fourth Amendment or in any other part of the Constitution. This is one major reason why the exclusionary rule has been so controversial in US society.
The exclusionary rule did not exist at all in the United States until 1914. In that year, the Supreme Court created the rule in its decision in Weeks v. United States. That decision only applied to federal courts, though. It was almost 50 years more until the Court used the case of Mapp v. Ohio to impose the exclusionary rule on states as well.