Is there a way to check the internet speed that the RPi is getting from the Channels web interface?
I think you can test it from a browser on another device on your network.
http://x:8089/speedtest
Using the RPi IP address in place of the x
No, that only tests the speed from whichever device is running to web browser to the DVR server.
At the moment, I don't believe so. If you did a standard install, you could always install your distro's package for speedtest-cli, and use that over a SSH session.
But as for using the Channels–provided Raspberry Pi image, such an option does not exist.
Over the internet
For remote test use http://my.channelsdvr.net/speedtest or [abc].channelsdvr.net:8089/speedtest