I’ve removed everything but the scrollPane (that thing with ComboBox in it) from TestJMEDesktop. Then changed height of it to 100. Now after the first opening of the combo’s drop-down list, and then clicking on other element (i.e. scrollbar), the list is not cleared. After subsequent opennings it is cleared properly:
Another thing: when I apply the GridLayout(1,0) to the desktopPane the drop-down list is shown on the side:
The second problem may be a result of me not knowing Swing very good - but it works in not-jme Swing.
Is any of this known/has solution/is not supported in JMEDesktop?
Hmm, the second thing is not surprising as the popup is added to the desktop pane. Thus applying a layout to the desktop pane is like applying a layout to your screen in normal swing applications - which is not possible. Thus: don't do that! 
The not-removed/cleared popup is quite strange. Can you figure out (e.g. by putting a breakpoint into JMEDesktop.LightWeightPopup.hide) if it's not removed or not repainted?
It is removed, but it is not repainted.
I tried to figure out why, but I'm a total Swing newbie.
If you modify TestJMEDesktop so that it createsMoreSwingStuff right after creatingSwingStuff you'll notice that the popup stays on the screen only if comboBox was the first component touched. (So in normal TestJMEDesktop, where you've got to press "more stuff" button to show the combo, this problem does not occur.)
It seems that one is solved with the last repaint fix.