How is called a category with endomorphisms only?
How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?
Every category in which every arrow is an endomorphism is a coproduct (in the category of categories) of monoids (a monoid is a category with just one object). So a category with all morphisms endormophisms is a coproduct of monoids. I'm not aware of any specific terminology for it.
Clearly the category of all coproducts of monoids admits an inclusion functor into the category of all categories. The construction you describe (of removing all non-endo morphisms) describes a right adjoint to this inclusion functor.