Look at the last two posts I made. One shows the ratings on the 3.4 amp adapter and the other shows the connector which is at the end of it (the other end of it is a standard wall plug). There is nothing else which gets plugged into it.
I realize nothing else got plugged into THAT black box. I was only talking about the black box that came with the lights.

You would have to use a multimeter to figure out which of those five pins were your power leads. (I personally wouldn't mess with that one just yet)
On the box, it states the 110V adapter is not included in the box.
Now there is the missing piece of information. Wouldn't it be nice if they at least showed you what kind of adapter it was supposed to be?
However, if I take the two wires from my 1.2 amp adapter, then would it matter which wire touches the cigarette adapter tip and side metal parts?
If it were
JUST a light, then I would say it wouldn't matter.
But since it's running through that box I would verify which one is positive and which one is negative on the two wires from the power supply first. (with a meter)
Typically the tip of the cigarette lighter part is the positive and the side tabs are the negative.
Also, how will I know whether the 1.2 amp adapter works. Ok .... obviously I know if the rings light up then it works, but do I have to worry about the rings getting too much electricity and starting a fire or something?
They will only DRAW as much power as they need. (current)
So as long as the voltage is correct, then just make sure the power supply isn't getting unusually hot or anything. And keep the fuse inline between the power supply and the light box.