Analyze the following sentences below about the excerpt of ...

I. In the structure “Oprah was deeply invested” the word “deeply” is as adjective.
II. The structure “the perfect person for each role.” is a nominal group connected to preposition “for” and the headnoun is “role”.
III. The word “invested” can be replace by “take part”.
IV. In the expression “on the casting choices” it refers to actor or actress in a play or film means to choose them to act a particular role in it.
Which ones are correct?