Answers:
No. Each enterprise can have only one enterprise program subscription; different departments within the same organization will need to coordinate with each other. After reading the comments, however, I'd say that "department" contains some ambiguity. I'd be surprised if the Department of the Interior and the Department of Defense have to share a single enterprise program subscription, but the Lingerie Department and the Footwear Department will probably have to work together.
No. There's just one team agent per subscription, and only the team agent's certificate can be used to sign apps for distribution.
No, there's no limit to the number of apps that can be signed with a single certificate, but I believe there are some restrictions on to whom you can distribute the apps. Read the developer agreement for details, but expect to distribute only to people connected to your organization.
Edit: I should add that the information above, particularly with respect to (1), is my understanding of Apple's policy based on conversations with ADR folks from a couple years ago. I have no reason to believe that the policy has changed, but the details of your situation differ somewhat from mine. Your best course of action would be to contact Apple Developer Relations yourself and get the most up-to-date answers for your situation.
