The events are written only to the calendars that are selected when you create/modify the event. If you want to have the same events in your local drive and in the flash drive you need to create two calendars (in Options->Calendars) and select them both in the event dialog when you create a new event.
You can enable the tray icon from Options->Advanced->ShowTrayIcon.
There is no Linux installer so Rainlendar isn't added to the Application menu. I don't know it either how it should be done and I would assume it is done differently depending what desktop you have (KDE, Gnome). This is my first real Linux application so most of this stuff is new to me too