How can I determine which network adapter my browser is using? How can I change it?

Your browser isn't choosing which interface your traffic goes through, the network settings in the OS are responsible for that. In this case it would probably depend on what the default gateway was set as. By default all the traffic would go through that unless there was an exception.

If this were linux the answer would simply be 'iptables' but in windows it's a tad more murky. I wish I could answer better, I'm sure someone else will have a better answer, but this should at least get you started.

You can use ForceBindIP to force applications (e.g. web browser) to bind to a specific network interface or IP address.

You cannot set the browser's default interface, but you can change Window's default interface. See this article from http://www.howtogeek.com/howto/27994/how-to-change-the-priority-of-wiredwireless-network-cards-in-windows/