Peer-to-peer sharing the way Napster worked only makes sense when you have lots of people sharing files. It's a bit of a chicken-and-egg problem, but it all comes down to this: no-one uses a network no-one uses. It's the same with social networks et al.
That being said, Qt4 does include a Network module which supports UDP and TCP, the basis for communicating over the Internet. There is documentation and example code available here.
As for your last question: I think the future for P2P is full of pain and suffering. Big Content is after the most-used 'feature' of p2p networking: sharing music and movies. Your opinion may be different.
