You would use the green terminal or not depending on what you want and what would happen if you use it or don't use it. For example if you have earthed PC connected with USB to a device that is powered by earthed power supply. If the device needs 10A current and by accident power supply black lead disconnects, the 10A current will find a return path via USB and PC motherboard to earth wire and via earth wire back to power supply. Most likely that damages something, and this would not happen if the 10A was provided by a battery or floating supply. Sometimes you want different sensitive devices to be or stay at known reference potential when you disconnect and reconnect them, so that floating potential differences don't break sensitive IO pins if ground disconnects last or first.
1) Yes correct. When there is black red and green terminal, it means that while you get e.g. 10V between red and black terminals, the voltage between black and green could be anything and can be set by for example the load. So output is isolated or floating from any other reference like earth. So basically the same as using batteries, they have no earth reference.
2) It just means remove the connection between green and black (or green and red) so the output has no reference to anything else. Some power supplies like PC power supplies have the black grounds connected internally to earth so it can't be disconnected by user.
3) Indeed if you have isolated power supplies (or at least one of them is) you can use them like batteries and connect them in series to make 20V out of two 10V supplies. But as the PC power supplies mentioned above have their black wires earth referenced, you can't connect two of them in series, as connecting 12V of one supply to 0V of other supply would create a short from 12V to 0V via the earth wire. Kind of like batteries having their negative leads already connected together, you can't put them in series any more.