I found this on one of the three main Stylesheets reference pages:

Note: With complex widgets such as QComboBox and QScrollBar, if one property or sub-control is customized, all the other properties or sub-controls must be customized as well.

This might explain why your arrow disappeared when you customized part of the combobox but you may have already known the nature of the quirk.

Fastest way I can think of would be to programmatically save out the icons you want via the non-static QStyle::standardIcon(QStyle::SP_ArrowDown), etc. With those image files on your drive, you can compile an external Resource that your stylesheets can access at runtime (assuming the resource syntax holds for stylesheets).

You may not be able to drop those "SP_" enums right into the stylesheet code but "url(:/qtStandard/downArrow)" seems reasonable.