Thank you so much, again, Mary! I worked with the links you provided, and ... I still didn't succeed. I'm afraid I don't have the time it would take me to try and err my way to success.
Luckily, I found a CSS compromise, which I'll share here in case someone actually likes it: I'm including @media entries in my CSS to reposition the header elements according to changes in screen resolution. If I couldn't remove the dock, I could at least make way for it! This way, there are at least no elements overlapping during extreme zooms, and it looks pretty good.
If you're interested there's a (temporary) draft at http://noodle.thereitis.org.