I believe so… 
If by dialog box you mean the dialog boxes created with the G4P.showMessage method or one of the G4P.select???methods then they use either Swing or OS dialog boxes and this has been covered in this discussion so I have nothing to add.
If on the other hand you are talking about your own dialog boxes created with GWindow then which icons are you talking about?
Which event handler are you talking about? Why do you need a different event handler when the window is invisible? Doesn’t make sense to me.