I've already set hotkeys of my liking for unit rotation but I wish there was a hotkey that did exactly what the bottom right round button does - so that I don't have to drag my mouse down there and click it for every unit / event that is in its list.  Is there a hotkey that does the same thing as that button?

It is in Settings, Keyboard and Mouse, and it is set by default to the ENTER key in the Numpad... you can set it alternatively to whatever you want, including a mouse button if you have one that has many more than standard. ;)