Yes, all of this is true. Other examples of 'categories of algebras of a monad on Set' include vector spaces, rings, modules over a given ring, representations

Yes, all of this is true. Other examples of 'categories of algebras of a monad on Set' include vector spaces, rings, modules over a given ring, representations of a given group, and Lie algebras. So if we could somehow teach category theory first (which would be hard because students would have no examples) we could shave a few lectures off each of those courses.