I don’t have any detailed knowledge about the Magnet specification itself, but until somebody more knowledgeable of the details comes by, I’ll give a quick answer, mainly to your second question.
Your quoted FAQ gives a very simplified explanation of what functionally happens, so unless you’re interested in protocol specifics that’s all you need to know.
Now, about the safety. The Magnet links includes a cryptographic hash of the torrent (or important torrent contents at least), which your client uses to verify that the contents have not been tampered with. This means that the torrent you receive from a peer is guaranteed (to an extremely high probability) to be identical to the one referred to by the source publishing the Magnet link. Of course, that only means that the torrent is as safe as the source which published the Magnet link, the actual downloaded files can still be “unsafe” as usual, but the files can’t be tampered with by peers without your client discarding the information as invalid. The same concept applies to all files downloaded even through “traditional” torrents.