Warning: I have not tested either of these plists yet and won’t be able to for few more hours 
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Try placing this plist in /Library/LauncDaemons/. Then either reboot or run sudo launchctl load -w /Library/LaunchAgents/net.duplicati.server.plist. This should start the server as root and make it accessible from the usual URL.
For the Tray Icon, you’ll need this plist in either /Library/LaunchAgents to run when any user logs in or ~/Library/LaunchAgents/ to run when only a specific user logs in. You may need to change this line:
<string>/Library/Frameworks/Mono.framework/Versions/Current/Commands/mono</string>
if your mono installation differs (check by running which mono in Terminal)