4.
Also, I know it's not the point of your request, but I'd recommend against having the search box left of the tabs. IE 11 does that and I keep clicking on it thinking it's the first tab I've opened! Or maybe I'm just an idiot. 
That's what's already implemented. I guess you never tried tabs bar in caption bar mode.
So it is! I don't use that view because it forces the MusicBee menus under the one button which means more clicking (and it isn't possible to select tabs with the cursor at the very top of the screen).
It seems arbitrary for the search box to be on the left for that view and on the right for others, but I use the Dark skin so at least the search box is a different colour which makes it easy to distinguish.
BTW, the key idea is not entirely new. This is how I use Chrome with bookmarks toolbar only with icons:

Once you try this, you will see why this is better than auto-hiding (if such option is available) or constantly showing bookmarks sidebar.
Obviously there are millions of applications that have toolbar buttons, but that's not the same thing - they're only acting as substitutes for a single menu item (or web site in the Chrome example). Buttons would work in MusicBee for some things currently in the left panel (like switching between the library and a radio station - hinted at in point #3 in my previous post), but the left hand panel is more than just a substitute for a menu. In your suggested layout, how would you browse to a folder in the Computer node and drag tracks to a playlist? Or drag a playlist to an external device? These are common activities which don't seem to have intuitive equivalents if the nodes were replaced with toolbar buttons.
I created this mockup to show how the new interface could look. What do you think? (click for full-size image)
This looks nice and would work fine if all you're doing is selecting one playlist to listen to at a time from a limited number of playlists. However, it doesn't scale well, either in functionality or number. If someone has dozens of playlists grouped in folders, selecting them from a drop-down menu quickly becomes inefficient. Also, additional functions like renaming, editing, shuffling etc. (all the stuff currently in the playlist context menu) would require displaying the menu first to find the playlist you want, then right-clicking that playlist menu item to display the context menu. It's not unprecedented for a menu item to have its own context menu (I see Firefox does that with its bookmarks menu) but it's not ideal.
Bee excellent to each other...