Both, like any Bittorrent client. The difference is it uses a library which prioritises certain file chunks so that you can watch it as you download (stream it).
Not really. Popcorntime doesn't provide copyrighted material, it just helps access and consume it, same as your laptop, TV, Chromecast or cellphone. Besides, the technology behind Popcorn could be used to access open content, therefore, the Berne Convention doesn't explicitly apply here.
I wonder if it'd be difficult to add an index from vodo.net to Popcorn Time. If they also added a link to the pay/donate API, that might actually give some of the Vodo.net-releases some much needed money (given the market share Popcorn Time apparently has).
For instance, here's a google search [0] that gives you access to the same pirated material that Popcorn Time does (but with a worse UI)
Does this mean google is an illegal site? Is HN an illegal site? It can get pretty complex.
[0]: https://www.google.com/search?q=yify+jurassic+world+filetype...