For an example of an impossible task, try to retrieve request headers using Selenium. Browser automation getting more and more complicated the more cases you're trying to cover and in my impression WebDriver is definitely not enough. Who knows, perhaps some new version of WebDriver that I never heard of it will catch up once the functionality gets properly defined.
Given that the selenium leadership is apparently uninterested in improvements, and it's many limitations, trying to improve there is more effort then it's worth.
Basically, they're still stuck in this idea that they're ONLY for emulating user-available input (and the dev-tools don't exist).
In reality, there is tremendous interest in more complex programmatic interfaces, but apparently they're unwilling to see that, and are instead only interested in their implementations "user-only" ideological purity.
Selenium can't do that itself but Selenium can drive a browser that uses a proxy and you can retrieve everything about the request from that. It's a lot more challenging if you're testing things that use SSL but it's not impossible with a decent proxy app (eg Charles).