I noticed on my combined flair that I had been awarded another silver badge. How can I determine on which site this badge was awarded without logging into all sites and looking for the inbox notification?
You don't have to log into all of the sites. Only one of them. The Stack Exchange™ MultiCollider SuperDropdown™ has a notifications tab that will show you exactly what new badges you have and on what site they were awarded -
As you can see from the screen shot, my two most recent badges were awarded right here on meta. You can see from the sites logo and also by clicking on the notification, you'll be sent to the relevant site.