American here...

1. Which wires go to the light fixtures?
2. Does the switch with wires #5 have two lights connected to it?
3. Is the switch with wire #3 illuminated?
4. If the white wire were disconnected, would the switch still work?

thanks for the reply:

1. not sure

2. the switch associated with wires 4 and 5 has one switch but two lights.

3. Not sure what you mean

4. white wire doesn’t affect the light/switch working normally (this switch is for a light that has two switches - one at each end of the hallway).

Ah, the white wire is called a "traveler". It connects two "3-way switches" together to allow them both to control one light. See the Option #1 diagram; the traveler is the red wire.

You can't use that DETA device using only the red wires you labeled because the DETA requires both the hot and common (aka neutral) wires. As shown in the Option #1 diagram, in America light switches control the hot (black) wires. Perhaps it is different where you are because you said that the red wires were labeled common. The black wires in the photo may be the hot wires. The green wires are the ground wires.

I'm guessing that the DETA device expects the common wire to be attached to the "N" terminal; the live, hot wire to be attached to the "A" terminal; and the hot wires to the lights to be connected to the "A1" and "A2" terminals. However, this is the opposite of the way the lights appear to be controlled now. Also, the DETA device doesn't have a terminal for a traveler wire so it can't be used in a 3-way circuit.

-- amateur