Is there a way to find out which app on an iPhone just made a sound? My phone is making two sounds, but not providing any other notifications.  I have no idea what is causing this.  How can I track this down?

I have it paired to an Apple Watch, and have Google Inbox installed (but not Gmail)
You should check your Notifications settings (Settings -> Notifications) and see which apps have "Sounds" turned on. I also recommend restarting your phone.