In KDE KWin can do things like that. Bring Firefox window to the front, then hit Alt+F3 and on the menu select ‘Advanced’ > ‘Special Window Settings’. At least here you could make it so that (certain) Firefox windows are initially minimized.
And for the System Tray, you need to create/modifiy an application starter, where the option “Place in System Tray” is checked. In ‘kmenuedit’, for instance.