Wiring Emergency Stop button to disconnect two independent circuits

The circuit will continue to function at reduced voltage. Note that the voltage declines over time. It does not suddenly drop to zero. Normally, you want to remove the load from the batteries before any one cell drops to zero volts, because the other two batteries will force the dead one to discharge to negative cell voltage. This "reversed" cell may then leak electrolyte in the battery compartment (depending on chemistry and construction details). If you plan to do this, you should consult with the cell vendor to make sure it will not cause leakage or other problems.

Most of my experience is with 1.5V alkaline cells and rechargeable Lithium polymer cells, not with button cells.

You can also test it by putting a discharged cell in series with two strong cells, then discharge the series stack until the weak cell reverses.

It's not only okay to connect the grounds together, it's mandatory if you want them to communicate or interact with each other in any way. Without a common ground reference, what you have is several entirely separate systems, and any communications between them would have to be isolated using something such as an optoisolator or a transformer.

Charging one set of batteries when they all share a common ground is also no problem.

That said, you really don't need separate batteries for different parts of your system, whatever the motor controller manual says. At worst, you will need separate regulation for high-noise parts of the system, and input filtering (such as LC circuits) to reduce noise from the motors. Just look at something like the Tesla- you don't see that having several different battery banks for different parts of the cirtuit.