Do I have to inform the Tribunal about my appeal to the Administrative Court?

Yes, you must tell the Chair of the Tribunal from whose decision the appeal is brought that you are appealing against the decision – see Civil Procedure Rules Rule 52 and Practice Direction 52D, paragraph 3.4

You must send your Appellant’s Notice marked for the attention of the Chair of the Tribunal which heard your case to the SDT by email to enquiries@solicitorsdt.com