Can I connect ground to my ungrounded porch?
Lighting branches didn't require grounds back then, so that probably explains why your closet was left ungrounded (I'm assuming there's no receptacles in there). My house was built in 1987 and I've fought a few ground battles on lighting circuits as well.

There should be no issue connecting your grounds. It's possible that your ground's continuity is broken somewhere else along the circuit that you're unaware of, but this would make you no worse off than you already are.

Once you connect your grounds, go back through and use a multimeter or a receptacle tester to verify the grounds are no longer open.

You can certainly use a wire nut, or some other acceptable means, to extend your short ground wire. This is called a pigtail and is quite common. If you do this, don't stop at a couple of inches-- go ahead and add enough so there is about 6" of ground wire as measured from where it enters the box, or where there is at least 3" extending past the edge of the box.