I knew I needed a different pair of eyes! I was sure that would fix it, but it didn't!. I'm noticing setting a breakpoint when the map is created, it's height and width are defined as 400x400 px. In the documentation, it says the size is determined by the div it's in. but since my div is within a floating pane that isn't yet opened, maybe it's getting confused? I have set explicit dimensions (comparable to the dimensions of the floating panel), to 100% to auto and to inherit (which all sound the same to me, but I'm sure have subtle differences!). Nothing is making my map budge from it's 400 x 400 size. In Firebug, I can set the dimensions of the mapDiv and mapDiv_root, which seems to be the container within it once it's renderer, but adding dimensions for the styles of those doesn't make any difference.
The map is still not resizing when I resize the pane. I have the resizeHandle.css referenced, which seems to be a common error people make. My 'workaround' is to have the dimension of the floating pane to also be a square, so it's not so obvious I have a problem. But that's no kind of solution!
I did finally dig around enough in the style of the dock to get my button to look better!