Hi, yes it doesn't have anything to do with timezones, not very plausible idea but just to rule out things..
Today is 14th and the tray icon still shows '13', I've left my computer on all night.. The calendar "window" shows the correct day (marked with the red-circle - standard skin).. About your questions:
-If I select Refresh in the context menu nothing happens!
-If I change the date, forward or backward (tested with some minutes and some hours) and I wait like 15 seconds after just in case -> nothing happens.
-If I change the date to Monday 15th, one alarm windows shows the events for 'today' but the tray icon doesn't change.. I changed it to Saturday 13th and then again to Sunday 14th and still the tray icon is freed. I have to say that the red circle in the calendar was updated correctly each time.
-I've just tried to hibernate and de-hibernate windows and still nothing happens..
-Finally I re-started Rainlendar and now a beautiful '14' is shown in the tray icon..
No idea :-/