Does ClearCaptions change any of the other services I have on my phone, like caller ID or call waiting?

No. ClearCaptions has no impact on these services.
I previously created a ClearCaptions account but now I want to sign in to it using Connect with Facebook.
You can only use Connect with Facebook for first-time ClearCaptions account setups. If you have a ClearCaptions account already, you will continue to use the account username and password you established when you signed up. You will not be able to log in to that ClearCaptions account using your Facebook login credentials.