Yes, downloading is akin to making a copy without consent of the copyright owner. (Edit: At least according to the Berne Convention, which most countries have signed.)
Edit2: I think dchest is right and illegality of the act of downloading does probably not follow from the Berne Convention, sorry. Distribution (offering for download) certainly is and I think at least according to German law the resulting copy has to be destroyed. Mind you, I'm no lawyer and might be mistaken.
I'm pretty sure Berne Convention says nothing about the act of downloading. Your use of "akin" may be incorrect. Of course, different counties have different laws, but if we try to infer legality or illegality of downloading purely from Berne Convention, I'd like to ask this question: how can you make a copy of something that you don't have the original in the first place? Thus, it's more likely that the party who's distributing the work is making a copy, not the downloader.
Theoretically yes but usually they target distributors in preference to users. It's easier to get a verdict or judgment against the distributor because it avoids the question of "what is a copy" and how that applies to the digital space and also because people are less sympathetic to distributors. The thing with P2P applications is that everyone is also automatically a distributor.
Edit2: I think dchest is right and illegality of the act of downloading does probably not follow from the Berne Convention, sorry. Distribution (offering for download) certainly is and I think at least according to German law the resulting copy has to be destroyed. Mind you, I'm no lawyer and might be mistaken.