Q:

Can't get rid of fake floppy drive

In my system it says I have "floppy0" mounted and I can't get rid of it.  
It isn't a real drive of any sort, since my computer doesn't even have a disk drive, when I click on it it says "unable to mount floppy0:

A:

Restart your computer and go into its BIOS (press Del, F10 or F12 -- read what it says -- when you boot up) and under one of the first parts of it, you should be able to disable the floppy drive by setting it disabled.  
Save and exit.  Ubuntu will then stop auto-detecting it. 
Just as a quick explanation for this: Old floppy drives (not the USB ones) lived on amazingly unsophisticated buses.  There wasn't an "auto-detect floppy drive" like there became for hard disks.  You therefore had to explicitly state whether or not you had a floppy drive plugged in if you wanted to use it. 
Seeing as nothing bad (apart from the behaviour you're experiencing) happens when BIOS told there is a drive when there isn't, motherboard manufacturers played it safe and left the option on, assuming (I assume) that people building their own systems would inevitably forget to go into BIOS and turn the floppy drive on and select its type. 

A:

You can also blacklist the floppy driver module (handy if your BIOS doesn't have the option to disable the floppy drive like many modern laptops):
$ gksudo gedit /etc/modprobe.d/blacklist

Add blacklist floppy to the end of the file, and reboot.