If you switch off the cube, then disconnect it from the mains and reconnect it, there is no power on the sockets from the cube.
The last state remains. Thus, if the cube is switched on, then disconnected from the mains and reconnected, there is power on the cube's socket.
The remote control is frequency-bound to the Cube, as it works via a pairing procedure. This means that the remote control is specifically connected to a particular PowerCube to avoid interference with other devices. If you already have such a PowerCube, the second PowerCube should not affect the first as long as they are paired correctly.
I just tried it out: 6 metres is no problem if there are no walls in the way. Info: the cube does not necessarily switch on or off at the first press of the button, you almost always have to press it several times. But that doesn't bother me.
Yes, with the remote control you activate all 4 T13 sockets simultaneously. If you want to operate another device independently, you either have to "respect" the on/off mode of the PowerCube Remote and switch the device on or off accordingly beforehand or you look for an independent socket, e.g. possible with a 3-way wall socket without any problem... ;-).
According to the manufacturer, 25 metres.
You can switch the unit on/off through approx. 2 metre walls, but with approx. 50% less range.
If the walls are made of wood, you will not have this problem.