How can I represent myself in a small claims court hearing on a defedant motion for partial dismissal of certain co-defendants within the group of an Insurance Company, where the names of said co-defendants has been used together interchangeably in communications by the primary Insurance company?

First if they have an attorney, you should not, but should have your own attorney, an attorney will know tricks to win, no matter what facts are.

But you take the communication with you, all of the written documents on their paper work, signed by them, to prove your case

Are you the plaintiff or one of the defendants?

Frankly, this looks too complicated for small claims court.

I am the plaintiff.

This appears to have some complicated issues of law which are unusual for small claims court. I would suggest that you consider filing a pre-hearing memorandum in which you set forth your view of the law concerning these issues.

When you go to the hearing,

Treat everyone with courtesy, particularly the judge.
Address the judge as "your honor" and rise when speaking to the court (unless told not to).
Wait your turn to speak. Never interrupt someone else when that person is speaking.
Allways address the judge, not the other parties or their attorneys (except when questioning witnesses).