Is AF supposed to automatically restart the application after changes are made to the settings in the control panel?

Active Forums never actually restarts the app. When you create a topic property it will add an entry to the shared resources file which triggers a restart from asp.net.

Do you know what if anything controls when ASP.NET gets this trigger? It seems to me the time lag is sufficient enough that doing a manual restart is faster on my site.

You can't control it. It happens automatically when a change is detected in the bin or app_localresources directories.