Does an initial algebra for a class have to belong to the class itself?
Yes, the initial algebra is by definition one of the members of the class for which it is initial.

You may however be interested in the category-theoretic concept of a limit. Given a diagram (a collection of objects and morphisms between them), we can ask for an object which "best approximates" the diagram, but needs not be an element of the diagram. There are two such "best approximations", namely limit and colimit. The limit is like an initial algebra.