How do 5 light switches in 2 boxes with all black wires work?
Yes, this is correct. In fact this is the preferred way to wire light switches.

One cable brings ground, neutral, and live in, 
live goes to the switch then a different cable takes live, neutral, and ground up to the light socket.

The white neutral wires are not connected to the switches because mechanical switches do not need electric power to operate.

for the new switches connect the ground wire to the copper wires. the copper wires should also connect firmly to the metal electrical box, there should be a green screw.  all grounds go to the same place, so allways connect all grounds together.

The white wire should connect to the white neutral wires in that wire nut.

The red and black will be a little tricky. the black goes to the black on the supply cable and the red to the black on the lamp cable, so you'll need to determine which is which. eg using a voltage detector.