What are some of the different interactions with empty remote servers in Android: Netrunner?
A remote server requires at least one piece of ICE protecting it or at least one card installed in the server itself (i.e. agenda/asset/upgrade) to exist. The three central servers always exist. &quot;Empty&quot; would refer to a server with only ICE protecting it, and nothing in it.

Empty servers already exist so is different from a new server. Cards like Mushin No Shin, which specify a new server, must interact with a new server (i.e. that they create).

If a server ceases to exist during the run (e.g. as the result of a Jackson Howard removal that you describe), the run immediately ends, and is considered neither successful nor unsuccessful.
The same is true if the trashed card is an ICE. For instance, if an empty server is defended only by a Data Mine which triggers, the server ceases to exist.