Years ago I saw someone using this (I think it was actually Yakuake) and ran it for a bit. I really didn't like that it was a single session. However, I later messed around with tiling window managers (which relates to your #2!) and having a shortcut to pop up a new terminal is really what I wanted this whole time.
Just to touch on #3, any time I'm writing a gui that's expected to have more than a handful of items I push to add a search bar with fuzzy matching at the top. Otherwise, search would get added after it's used for like a decade and everyone is immediately enthusiastic and grateful. All of the non-fuzzy solutions are too pedantic to use.