OK, first of all, plainly this is a mess. The fact that there is a white wire hooked up to a black wire without any recolouring of it alone says that (1) there's some amateurish stuff going on here, and (2) you cannot trust any of the white wires to actually be neutral until you prove that they are. So proceed cautiously, and take notes as you go.
There are a lot of questions here but a good one is:
What's the easiest way to track a specific wire to see how it flows without ripping out the walls?
Start by obtaining a long three-prong extension cord. Plug it into an outlet that is on, and verify with a voltmeter that there is voltage between the hot and neutral, there is voltage between the hot and ground, and no voltage between the neutral and ground. You now have an extension cord which you can use to provide a known ground and a known hot.
Next, turn off the power to everything that you're going to be working on.
Next, put a piece of tape with a number on every wire that you're going to be working on, and start making notes about the color and location of each numbered wire end, and what they're hooked up to now. Take photos. You want to be able to put this back together the way you found it later. Number the switches as well.
Next, carefully get all the wire ends exposed but not touching anything, and turn the power back on. Use your known ground and your voltmeter to determine which wires are hot under every combination of switches. Remember, some of the whites are hot.
Turn the power back off again.
Test everything again to make sure that no, really, the power is off.
Now test everything again against the known hot to find out which wires are neutrals that run back to the panel. Again, test everything against all combinations of switches, remembering that at least one mistake has been made in this wiring so far; there might be more. Interrupting a neutral with a switch is almost always wrong, but I have found plenty of interrupted neutrals in old houses.
Now unplug your extension cord from the wall and get a flashlight. Rig yourself up a continuity tester out of the extension cord and the flashlight. (Of course if your voltmeter has a continuity tester, use it instead of jury-rigging one out of a flashlight.)
Suppose you suspect that wire ends labeled 2 and 9 are the same wire. Attach end 2 to the extension cord ground. Run the extension cord to the other. Now wire the extension cord to the battery, the battery to the light, and the light to end 9. Did the light go on? Then your hypothesis is confirmed. Did it not go on? Try every combination of switches. Again, record your observations.
Keep doing that until you have enough observations to form a consistent theory of how the wiring runs through the walls. Draw a diagram.
Once you're there, you should be able to solve your problem. If you cannot, then you'll be able to post a question here with a lot more information.
Also, take this opportunity to ensure that every wire is correctly coloured, by putting tape on miscoloured or confusing wires. Also, if there are oddities such as a box that has hot wires from two different breakers, make a note of that for the next guy.
Based on your readings, the 1 side black, 2 side white(! retag this wire with black electrical tape!) and 3 side black should be nutted together and pigtailed, and the 1 side white should be pigtailed to the 3 side white apparently -- there looks to be some sort of load plugged into the outlets connected to the 3 wires, which'd explain why the 3 side white reads 120VAC with regard to the 1 side white. From there, you can then connect the white pigtail to the silver screw, the black pigtail to the lower brass screw, the 2 side black to the upper brass screw, and the bare pigtail to the ground screw.
The funny voltage readings on 2B with the switch off are due to the high impedance of your meter seeing capacitive coupling of a small amount of energy into the switched-off wiring.
Best Answer
Breaking the tab on the neutral side does exactly the same as the hot side: it separates one terminal and neutral outlet from the other terminal and outlet.
I can't imagine a scenario where only the neutral would be cut.
The neutral should be cut where the corresponding hots are cut and come with their own neutral. For example, if the two halves of the outlet are controlled by two separate switches and the power comes from the switches, then the neutrals should be separate. The neutral should not be cut where the power comes in to the box with the outlet, and the wire from the box to the switch is switching the hot and returning it over the white wire.