Hi Bruce.. I'll add this to the other advice
1) Am I correct in assuming that the voltage should be across the black and white wires (within one cable), and the lack of this probably points to the problem? (And is the voltage across black and copper normal?)
Yes you are correct also 120 volts black to bare ground is what you should be getting. This also proves you have power to the receptacle.
2) What problem am I likely seeing?
O volts black to white is telling you that you have an open in the white wire or commonly called an open neutral as househelper and Mike have already stated.
If you determined that this closest to the panel outlet receives power first before the others then your open white is upstream of it as you are measuring O volts black to the white wire here. But you need to make sure you have the incoming power cable isolated when you test it. All power should disappear from the other recepatcles if they are daisy chained to that outlet and with the cable isolated its black should be constantly hot and measuring 0 volts to the neutral of that cable.
Would this be what you have determined?
If so the problem is not in any of the non working receptacles so you will need to see what is working on that circuit and look in the switch box or light box for a bad connection or open with the neutral white wire. Remember while your looking not all whites are neutrals and can be used as hots...like in switch loops so don't let this confuse you. To find what else is on the circuit turn off the breaker and see what else stops working look in the electrical boxes of these areas for the open white wire.
If you have determined that this is the first device the circuit hits before going anywhere else then isolate the incoming power cable and test it.. if 0 volts between black and white then problem exists between there and the breaker. Now something to watch out for if this is the case. When you replaced the breaker did you notice if there was a red wire in the cable with the black when it exited out of the breaker panel? I'm asking because you may be dealing with a multiwire circuit and there will be a junction box somewhere you will need to find. I will pursue this further if that is your case...check for that red wire.