Part 1:
If the button has icons, you can do that using the label placement option, but buttons always have a bit of a margin/padding on the sides.
Even if you could turn off the margins/padding, that would not look good. The kerning would not be right.
You can already assign separate actions to buttons when the left, middle and right mouse buttons are clicked, which I think handles the aim of having dual-action buttons far better than this.
Parts 2 & 3:
Nothing to add that the other replies haven't said already, except: