I have a two prong outlet in my house and there is a ground wire behind it, not connected. The ground wire ends in a junction box halfway to the breaker box. Only the neutral and hot wire continue from there.
Can I connect another ground wire and continue it to the breaker box?
As long as the wire is sound and is joined old and new properly, i.e., not just twisted together but secured with a twist lock or crimp connector it meets code and is fine. Consider that the hot and neutral is already joined there so the ground connection does not degrade that circuit and is quite proper. That said - anytime you can replace old wiring with new is a good thing. If you are running a wire from the panel - I would recommend you run entirely new nm or romex cable as far as you can.
Yes you can, but the proper thing to do would be to run a whole new cable from the panel to that first box. If you can do it with a ground wire it is not much harder or more expensive to do it right with a new cable.
A ground wire needs to be connected to the ground bar in the breaker panel where the branch circuit originates. If you have only hot&neutral leaving the breaker panel, and then further on the circuit someone used Romex with ground to extend a circuit, you will need to run additional wire to ensure that the receptacle is truly grounded and not just wired to a ground that goes nowhere.
In some areas (check your local codes), a circuit can be grounded to a copper or iron plumbing pipe, using the appropriate clamps and connectors.
You can also test the receptacle using one of these guys, once you've got it wired up:
