Does Fastmail.FM have the equivalent of the "Archive" action in Gmail? Mail.app on OS X and on iPhone seem to play well with Archive and it's not something that I really want to give up.
No, and it's really annoying. I have an "Archive" mailbox in my Fastmail account, and Mail on both OS X and iOS recognize it as special (they give it the same "archive" icon as a GMail account), but don't override the 'delete' function. I haven't found a Radar for it, but I imagine that many of same have been closed as "works as designed".
Mail.app's "Archive" isn't the same as Gmail's. In Gmail, it just removes it from the current mailbox but leaves it in All Mail. Mail.app actually moves it to a specially-named IMAP folder.
Yes, there is an archive button. There is a difference, mail in Fastmail is in traditional IMAP folders and don't use tags as folders, I assume that you just tell the mail client not to download the archive folder.