one of the outlets is wire like shown with a wire going from the ground screw up to one of the main screws. Is this ok?
No, that's not okay. That is an ungrounded circuit and should only have a 2 prong outlet on it (or a GFCI outlet).

Another thing is that when we use a tester on the light switch, no voltage.
I'm confused. You said that you are changing out the devices because of looks. So does the switch work? If so, then you are just doing your test wrong (or you turned of the circuit!?).

A light switch won't show any voltage across it's terminals if it is ON. It may not show enough voltage for some testers if it is off.

As greg points out, the wire between ground in neutral is DANGEROUS and ILLEGAL. Your options are as he stated: Put a two wire receptacle in there or put a GFCI in there (in this case you must mark it and any downstream protected receptacles as NO EQUIPMENT GROUND).

That presumes there really isn't a ground available (though the guy who did that hack goes from criminally stupid to having the IQ of Jell-O if that was the case).

Thank you for your help.

We think the cable coming into the outlet does have a ground wire that was cut off short inside the device box.

Inside the light switch box it looks like that ground wire is pigtailed in with the other ground wires.

If there is in fact a ground wire that was cut short, can we try to get enough of it to attach to the ground screw on the outlet? Then would the 3 prong outlet be ok?

And yes, the light switch does power that outlet, it works.

If you can get enough slack to attach the ground to the outlet, then the 3 prong outlet is fine.
If there isn't that much slack, you may be able to attach another length of bare wire with a wire nut.

Note that the metal box is supposed to be grounded too. So the ground wire that is coming in has to be connected to the box and the outlet.

And if that switch controls the outlet, then testing across the terminals will always result in zero voltage, whether it's on or off, since there is no load on the circuit (ie. nothing is plugged into that outlet).

If you want to test for power at the switch, you put one probe on the black and the other probe has to touch the neutrals that are combined with the wire nut.

With the switch on:
Touching either screw and the neutral will show 120 volts.

With the switch off:
One screw and the neutral will show 120 volts. This is the feed.
The other screw and the neutral will show zero. The switch is breaking the circuit.

Awesome, thank you so much.

With the switch on:
Touching either screw and the neutral will show 120 volts.

With the switch off:
One screw and the neutral will show 120 volts. This is the feed.
The other screw and the neutral will show zero. The switch is breaking the circuit.

That's provided he has a neutral (though in most cases there's a ground which will work fine for this debugging purpose.

If this is an older home, then it may have been wired with BX. If that's the case, would it be OK to connect the ground wire on the receptacle to the metal box itself? I assume that if you measure the voltage between the hot wire and the box and get a solid 120 volts, then there must be a continuous ground connection back to the panel. Is this correct?

At the time it was installed it was likely OK to use the BX sheath as the equipment grounding conductor. Unfortunately, that was subsequently found to be a bad idea. The newer Type AC cable (which looks like BX) has an extra ribbon of metal connecting all the spirals of the sheath to make sure it's a good connection.

This house was built in the 1940s. There seems to be several different kinds of wiring used over the years.

We removed the paneling, etc from one room down to the studs and discovered this connection:

It appears that the black wire has no ground wire in it. The white wire has a ground wire that is just loose there and in the device box.

In the attic there is a bunch of this black wire being used. One cable has even been spliced together and covered in electrical tape. I think this particular cable runs to the breaker box.

Is any of it grounded?

I don't like it at all and want to replace it all. It is so old that it sounds like it is cracking when you bend it.

As suggested in a response to your other thread on this subject, it is both not to code and is not safe. For me, safe is the first priority and legality is the second. If it is not safe, the possible penalty can be swift and severe and imposed by the situation or equipment. If illegal, it must be discovered by the proper authorities for penalties to be assessed.
Getting your situation properly resolved with piecemeal posts and fixes is not likely. Get a professional to your premises.

There is no ground in that 2-wire black cable. Therefore there is no ground in the 3-wire beyond it. Someone may have put 3-prong recepticals afterwards but none of them are actually grounded.