I use Tree Style Tabs as well. As you said, vertical space is a premium.
Back in the day (and perhaps still), there were user chrome tweaks you could make to hide the address bar until you hover over it.
I am currently experimenting with 'Toolbar Autohide' (Firefox). It also hides the address bar and other tools. Meaning I now have the complete vertical screen space available for website content.
Vertical space is not really at a premium when it comes to viewing static documents under access to a scroll wheel.
Why does the developer console open on the bottom, constraining your vertical space? Because there is such a dearth of horizontal space that that it would be uncomfortable to do otherwise. The reason things keep eating up vertical space is because that's the only space that's not at a premium.
For me, vertical space IS at a premium. Especially when browsing the web. How many web pages do you see utilizing the full width of the monitor? Very few, which is also the reason I choose to dock my developer console to the right instead of the bottom. This also doubles as an easy way to resize the viewport when testing responsive design/media queries without resizing the entire window.
Most monitors are 16:9 with the 16 being the horizontal dimension. It's also difficult to read very long lines of text, so my browser windows tend to be narrow and tall. When I use the developer console, I pop it out into another window and put it next to the original window.
> Most monitors are 16:9 because horizontal space is more important than vertical.
Most monitors are 16:9 because that aspect ratio fits certain popular entertainment media content that people consume, and a significant usage of general purpose computers is consuming that content (and, also, a significant use outside of general purpose computers of displays is for consuming that content, which effects the economics of producing panels.)
That doesn't necessarily reflect the relative value of horizontal vs. vertical space in other applications.
With a 1080p display, my windows only use half the screen, and I hardly ever feel that it's too narrow. What pages are you browsing that benefit from being wide?