1. You could leave it as is, or you could write, "The ghost sympathizes with Gertrude." One is an adjective; one is a verb. They are not interchangeable.
2. Put a comma after "play," and use it as is. (Nothing in the parentheses.)
3. All three expressions are correct, yes.
4. Are you referring to "Vice" as one specific character? If not, then delete the quotation marks and use lower-case v.
5. Gertrude's imagination is working feverishly and may cause her harm.
6. He pleads that Hamlet show sympathy to Gertrude. OR He pleads to Hamlet that he show sympathy to Gertrude.
7. The ghost spurs Hamlet into action.