I need a mobile device to be able to scroll a large, static map image, and also be able to select PNG image "objects", like this:
What would be the simplest method? HTML5, jQuery, Sencha Touch, or something else?
I would suggest at least taking a look at the PlayN engine. I believe that it can support the type of behavior you are asking for, and it targets HTML5, Java, and Android platforms (with work currently in-progress for targetting iOS platforms). This is the same engine that was used to port Angry Birds to ChromeOS.