WP Customizer – what types of fields is it capable of supporting?

The reference you pointed at just lists the built-in data input types. You can always add you own by creating a custom Customizer control. Follow Otto’s guide to create a textarea control. If you know your HTML, it should not be harder to implement a range slider.

I personally use the Customizer API to modify the background-position CSS property of the background image using the mouse dragging movement (see the git repo). Possibilities are endless !