Terminal doesn't force quit When I click Terminal in Launchpad, it says "The application Terminal is not open anymore." But it shows up in the Force Quit window and it has the open-app indicator on the dock.

I can't quit the app, restart or shut the computer down.

I think this could be the root of the problem:

I was running a program in Terminal overnight and then it said something like, "Terminal is paused...run out of memory." I quit it and when I opened it again, this happened.

Using 
MacBook Air
OS X El Capitan v. 10.11.2
Sounds as if you may have run your system out of RAM, which is quite possible with the anemic Air designs Apple is passing off as computers these days.

you may well need to do a hard shut-down and reboot.. hold down power button till it shuts off, then wait 5-10 seconds and reboot.

If terminal does the same thing after that, then you may have an actual issue..