As to your hard drive, no the file will not show up there... Most NICs don't run in permiscuous mode, so they'll only get the traffic that is addressed to them.
OK, I haven't messed with wireless in the specific, but I imagine that with some of the routing, there is a bit of a commonality here. Beneath the level that IP functions at (which is where you get your logical IP address), there is a layer in the networking model refered to as the datalink layer. What's important here, is the part refered to as MAC (media access control). Each NIC has it's own MAC address, which is physically tied to the network card...
When the router needs to access a given device, based on it's own routing table, it determines where to send it (in your case, out the wireless, or out the connection to your ISP, I'm imagining is your setup). It also grabs the physical address which associated to that IP. Because the MAC on your computer wouldn't be the same as on theirs, it would be addressed to their computer only. Nothing would show up on your hard drive, so no if the RIAA, law enforcement, or whoever came looking, there'd be nothing on your hard drive to implicate you with.
However, you should still secure this. Because, as you have already guessed, the ISP could have a way of knowing what's going over your connection, and the ISP could hold you accountable for this. If someone breaks into your WAP, and commits a TOS violation (and illegal activities likely do fall under your terms of service agreement), it would be up to their discretion to decide how such a breach of TOS they might be engaging in, will be dealt with. Once you know this is going on, if you fail to do anything to secure your wireless, they could hold you accountable for what knew, but allowed to pass through your network circuit with them, anyhow... Afterall, the router is your own internal network (your hardware), which is under your control; just as their network is under their control. A prob with your routers or other networking hardware, that you have added to your connection (was not provided by them), would be deemed your responsibility.
It would be adviseable to secure this, and if you need help figuring out how to secure it, to ask someone. It's also a way to cover your butt "I found out someone was doing X, and I have since secured my wireless, sorry..." vs. doing nothing at all, and allowing the activity to continue over your connection. But, your hard drive itself is safe...