The IP address you’re entering in your screenshot isn’t a publicly routable address. As such, there’s no way that particular address will work outside of your LAN.
You don’t need to use a third party app for a reverse proxy, but it’s highly recommended for easier security with certificates and not needing to specify a port number in your URLs.
Plex and Emby don’t strictly need a static IP for your local server because they have centralized authentication that you bounce through. ie: logging in through app.plex.tv or Emby Connect don’t go directly to your server, they go through their hosted servers first. At a minimum, you need to have a publicly routable IP address and to open the appropriate port(s) in your firewall to allow traffic in.