Well, it's not really F3. It is the associated function on F3. The F-keys have two functions, one without the fn key and one with the Fn key. These can be swapped in the Keyboard System Preferences (use al F1, F2... keys as Function Keys). If that is checked, the F-keys are F1, F2, … and to get the pictured functions you have to hold down fn.
In Mission Control, you can change the settings for which keys are used for the functions. If you choose an Fn key, then that is the key. It is not the picture function attached to that key.
Mission Control used to use F9,F10,F11. However, that has recently been changed to ctrl-up/down arrow.
Are there any other problems with the Tab key? Does it function normally indenting text?
You shouldn't be able to set a shortcut to cmd-tab, but it might be worth looking through Shortcuts to see if you have managed to do that. You can also try resetting the shortcuts to defaults. However, my guess is something else is interfering with the behavior. Do you have any other input sources (Tablet, Mouse, Keyboard) that uses a kernel extension to interact with the OS.