Regarding your first question, I think the following definition and theorem settles the answer to the affirmative:
Definition: A coalgebra $C$ is called right cosemisimple (or right completely reducible coalgebra) if the Category $M^C$ is a semisimple Category i.e. if every right $C$-comodule is cosemisimple.
Similarly, left cosemisimple coalgebras are defined by the semisimplicity of the Category of left comodules.
Theorem: Let $C$ be a coalgebra. The following assertions are equivalent:
- $C$ is a right cosemisimple coalgebra
- $C$ is a left cosemisimple coalgebra
- Every left (right) rational $C^*$-module is semisimple
where $C_0=Corad(C)$ i.e. the coradical of $C$, which is the sum of all its simple subcoalgebras.
For a proof of the above you can see for example Hopf algebras-an introduction, Dascalescu-Nastasescu-Raianu, Ch.3, p.118-119, Theorem 3.1.5. (However this is more or less standard material, you can also find it in Sweedler's book on Hopf algebras, etc).
Now, regarding your second question, I think you can use the above theorem together with the correspondence between $C$-comodules and rational $C^*$-modules to investigate the situation closer.