Well, I have discovered one thing. It appears that my setSelectedValue() commands are working. If I put this code in a button...
var run = DROPDOWN_RUN.getSelectedValue();
APPLICATION.createInfoMessage("Initial value: " + run);
DROPDOWN_RUN.setSelectedValue("AUDIT");
run = DROPDOWN_RUN.getSelectedValue();
APPLICATION.createInfoMessage("Set value: " + run);
...I can see the value is getting reset to AUDIT, as I want. But, the DROPDOWN_RUN dropdown doesn't display AUDIT. It continues to display whatever value it has when I click the button. Weird.