Prions are considered infectious, although when they were first proposed it was essentially heresy that something could be infectious without containing any nucleic acid (DNA/RNA).

Whether they self-replicate or not is sort of a semantic issue. They replicate in the same way "fallen-down dominos" replicate.

Prior to being a prion, they are an already complex protein, but by folding a particular way, they become a prion, which causes other proteins of the same type to fold similarly.