It's generally a good practice to keep different software installations in separate directories to avoid potential conflicts and to make it easier to manage and troubleshoot. Installing a game server under the RAT install folder may cause issues with file permissions or other conflicts that could interfere with the server's performance or functionality.
Regarding the mods, it's important to only install mods that are proven to be compatible with both the game and any other software or tools being used. Installing too many mods or incompatible mods can cause issues and make it harder to diagnose problems.
Enabling/disabling SteamNetworking in the config is a good suggestion to try if the server is not showing up on the Steam server list. It's also a good idea to keep the server running for some time to see if it eventually shows up on the list, as it can take some time for servers to appear on the list for some players.
In general, when troubleshooting server issues, it's a good idea to start with the basics and eliminate potential sources of problems one by one, such as checking file permissions, verifying that required software is installed and configured correctly, and testing with minimal mods and settings to isolate the issue.