That's an awesome idea (using the inductor).
Now this brings up the question of safety. How can I make my inductor more safe. I'm aware of the possible 'complications' if the inductor, for whatever reason, were to make contact with the live 110 wire... YIKES ! How can I make this more safe ?
Well, there isn't any real danger so long as you don't break the insulation around the wires you are testing. It's not a great idea to put the inductor really close to the end of the wire where it has to be bared. Put it at least a few inches away and there really isn't a whole lot of danger. Standard wiring is rated up to 600V. I don't remember the dielectric breakdown voltage but it would practically take a lightning strike to do that. So, in a lightning strike you are likely to see your arduino blow up. If you want to prevent that you'll need optoisolators and then you'll definitely need an amplifier.
Also about amplifying the signal, how can I go about doing that, using another transistor ?
I don't think that an inductor would properly turn a normal transistor on. It's likely to be too weak. You'll need an amplifier IC. You can get them really cheap.
Also, I forgot, an inductor will produce AC voltage. You'd have to run it through a diode before it would be safe to run it to a pin on the arduino. Of course, most diodes have a drop of at least .3v so if the inductor can't produce more than that you'll never see it.
And about the 3-way relay wiring, I still don't get it but bare with me. I'll sit on it for a day or two and I'll figure it out, ha ha.http://www.misterfixit.com/3wayswch.htm
Go there. The picture is really about the clearest that the concept can be explained.