For future reference... can you get away with saying that someone was using your wireless router?
I don't think so unless you have hard evidence, and even then you're probably still responsible. It's like having drugs in your car but them not being yours, you're responsible since it is your car.
No, you're not. If someone else in the car claims the drugs, you're off the hook. And yes, I know this for a fact.
We're not talking about someone claiming them. Think of the analogy, if someone pirated something via your network they aren't going to be like "Yeah, it was me." Drugs are a different story because you're both physically there.
How did they notify you?
The company which the studio hires to monitor for infringement sends out an email to your ISP detailing the infringement and your ISP in turn emails you about it. With Cox they tell you that you have x amount of days to contact them and resolve the problem or they'll terminate your service. When this happened to me I made the mistake of emailing HBO's lawyers and apologizing (I did this from my moms account and said how I taught my son how this disrupts capitalism and such). The guy at Cox said that was a bad idea since now they actually know who I am, he said that most likely nothing would happen since it was for a TV Show but had it been for music files the RIAA is much more persistent and the chances of charges being filed would have greatly increased. Luckily nothing happened. Their main goal is for you to stop pirating their property, not to disrupt lives. As long as you comply then you'll be fine. (This is the problem with P2P because you're automatically an uploader and since you're sharing it that's what you get in trouble for.) You're much better off using private uploads on sites like Rapidshare or Sendspace. Or watching for additions on sites like quicksilverscreen
