No. You can split the hot feeds and you can split the neutral feed, but both outlets of a standard duplex receptacle share the ground.
Hot, neutral and ground.
the bare copper is always a ground
Yes, all receptacles require an earth ground. (the bare copper wire in house wiring). However, it is not a dedicated ground in that all ground wires in a houses electrical system are connected or in common.
It's possible that the outlet is on a switch, and either one half of the outlet is switched or the whole thing. The extra two pair of wires probably feed the NEXT outlet.
In North America (Canada, United States, Mexico) a household 110V-120V outlet has three wires: hot, neutral and earth ground. If the outlet is mounted in a grounded metal box, the third wire (earth ground) may not be present, since the connection to ground occurs through the metal outlet housing and possibly metallic conduit that routes the wires from the circuit breaker panel to the outlet. Where metallic conduit isn't used, the earth ground wire may be connected to the metal box, and the earth ground pin of the outlet is connected by securing to the box with screws.
You use a known ground and check them for voltage. You can use an extension cord to reach from a ground to the wires you are testing. You are not putting it in an outlet, just to connect you to a ground.
You need two different hots, one neutral and a ground. You can't do it with three wires.
If the wiring system into which you are installing an outlet has no ground available, use an ungrounded outlet. In an ungrounded system, an outlet with a ground contact would allow the outlet user to mistakenly, and perhaps dangerously, assume that a ground was present. A suitable ground may be available as a ground wire accompanying the hot and neutral wires in the cable, or a ground may be available via conductive conduit and a metal outlet box. In any case, use a tester to confirm the integrity of the assumed ground. A voltage test from the hot wire to the ground should show the same voltage as between hot and neutral (the black and white wires respectively). If you are replacing an ungrounded outlet, you need not assume there is no ground present. You may find, in the box, ground wires that were not connected to the outlet. You may come across grounded outlets that have no ground wire attached because they rely on grounding via the mounting screws through the outlet ears to the metal box. This is a less reliable grounding method. It is better to buy a ground-wire "pigtail," fasten the wire directly to a hole in the metal box with the supplied screw, and attach the other end of the ground wire to the outlet via the outlet's ground screw.
In a standard outlet 3: hot, neutral, and safety ground. In an appliance outlet there may be 4: hot #1, hot #2, neutral, and safety ground. In an industrial 3 phase delta outlet 4: phase A, phase B, phase C, and safety ground. In an industrial 3 phase Y outlet 5: phase A, phase B, phase C, neutral, and safety ground.
I consider a connecting wire to be, when you have power coming into a box and leaving, the neutrals, and hots are tied together, then a connecting wire is added to wire to the outlet. Anytime wires are put together, but one is added as a branch off of the wires tied together. Most times the wires just get attached to the outlet, but you must use a connecting wire for the ground to go the the outlet and one goes to the box,unless it is a plastic box. You should not really just wrap the ground around the ground screw and then to the outlet, it should have two connecting wires added to the ground where both are connected together, if there are two sets of romex, one entering and one leaving.
If you still have a non-grounded outlet. One that does not have a ground wire you should replace the wire that feeds that outlet and not just add a ground wire from another source. The main reason is your feed wires are over 50 years old and could have other problems
The green ground wire should be attached directly to the junction box. That is the metal housing where the house wires enter the outlet.