You can try the 2.38.1 beta version available in your account. That version has a default icon setting in the MapPress settings screen. The default in this case is the icon to display when no specific icon was selected in the map editor (i.e. instead of the ‘red dot’).
Icons-by-category is on my development list but I have no date for it yet. However, if you build your maps using PHP template tags, you can set the ‘iconid’ property to whatever you like. There are some simple examples of template tags in the documentation.