That’s quite a load of topics 
Your server configuration is part of the account with which you setup the server – so if another user is logged in and runs the server, they won’t see what’s on your server.
IIRC there’s ways to run the Plex Media Server “as a (Windows) service” in which case it will not depend on a particular user to be signed in.
Remote access is a book of its own. There’s many factors that can impact your setup – a VPN will certainly complicate things. You might want to take a look at this guide to pin down what might be causing this (e.g. if it’s still not working if you bypass the VPN for the Plex Media Server).
As a side note: you don’t need remote access to stream media inside your home network.
As for the app not linking to your server… on what platform are you experiencing this (e.g. Samsung Tizen, LG WebOS, Android TV…)? Certain older TV models are no longer able to connect to a server securely as they don’t support current certificates.
Usually the Plex apps on a TV will ask you to link the app to your Plex account. Usually this works by the Plex app offering a 4 character/digit PIN and a URL → https://www.plex.tv/link/ to enter it. Opening the URL in a browser on a mobile/PC will ask you to sign in and enter that code.
From what you describe it sounds like you’ve enabled the DLNA server feature on your Plex Media Server and your TV has an option to use different DLNA sources as inputs
