I'd separate the year and month and make the year controls visible.

But all of my UI design work is targeted at internal tools where looking good is way down the list below ease of use, so take my opinion with a grain of salt for anything customer facing.

Ex: A down arrow button attached directly to the right side of the year input field. Make it a deeper blue and then wrap a 1 or 2 pixel wide rectangle around the field itself of that same color.

Well, the web 1.0 way to indicate that text was clickable was to make it blue and underlined. What was so wrong with that convention?