How would I go about calling a random public thumbnail and link from a site on another server?
accessing from another server would mean having mySQL access from another server - usually only localhost (your server) has access to the database. You might want to look into the rss mod or the random image include mod though (both of them in the mods board), which will let you create one or more random thumbnails on a custom php page on your server. You could then pull the content of this file from the remote server (using php includes or whatever technology you can think of).
GauGau
Use RSS mod