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.
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.
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.
Actually, the developer console has defaulted to the right side on new profiles for a while now.