First of all scan with an anti-virus.
Second:
#1: Control Panel > System > Advanced options > the first "Configure..." button > Enable "Show shadows under icons text on desktop"
Edit: You may also have changed something in the Icon interspace in screen setting, check it out if some folder names still override one line.
#2 See Tochjo's post
#3 Select the properties of the shortcut and set in the name "Internet" and in the comments "Firefox"
Edit: I changed to the modersn Start Menu and I found out that Firefox is automatically set there. Tochjo was kinda right
( right-click on taskbar > Properties > Start Menu > Configure... > General > Show on start menu > Internet > Firefox )
/ –
