UPDATE:
I was doing all of this on the BBB connected to a Mac over USB.
Now I have plugged my BBB directly into my home's Internet router using Ethernet, and ntpdate-sync and opkg update are working correctly. I'm able to ping 8.8.8.8, and all appears to be working properly.
So I guess if the BBB needs access to the network, it must do so over Ethernet only?
When I use USB to connect and ssh into the BBB, I'm able to browse to the BBB and it says that it's connected, but the above DNS problems result.