ISPs typically keep logs of when their DHCP servers assign addresses, and to whom those addresses are assigned. The RIAA can they say "Who was IP address 1.2.3.4 assigned to on 2/23/2008 at 03:57:24 GMT?". Their query likely would not be as lucid, but you get the point. If the address was changed very close to that time, the true lessee may not be accurately known because of time differences. Unless the DHCP server and the system making the query are synchronized (to an NTP time server, for example), it may not be possible to tell who the lessee was. But I suspect it's unlikely that many queries will match this time-too-close-to-assignment scenario.
As for why ISPs charge more for static IPs, it's because when they assign a static IP address to you, they cannot use it for anyone else. And just like you, your ISP has to pay for their block of IP addresses. If everyone is using dynamic addresses, it's not such a big deal because everyone will not be online at the same time, so the ISP can oversell the address space (much like high-speed internet [broadband] providers oversell their bandwidth by claiming "up to" excessively high speeds and then cutting off people who use those speeds consistently).
