Is there any way to send arbitrary headers? I’d like this to send a particular cookie value, but it would also be useful to set user-agent. (I realise you can also test via different browsers.)
Yep, you can add cookies, set the UA or add/modify arbitrary headers if you use a script: https://sites.google.com/a/webpagetest.org/docs/using-webpagetest/scripting
For future visitors, the updated documentation URL is: https://github.com/WPO-Foundation/webpagetest-docs/blob/master/user/Scripting.md