In mail app, What is the gray box with an arrow pointing down into it?

It sounds like your folder list icon. Clicking it should bring up a list of mail folders. If you were reading an email and clicked that icon it should have pulled up a list of available folders to where you could move the message. If this happened, click the icon, select the folder to where you have moved it, open the message, then repeat and move it back to the inbox folder. If you know what the message contained you should be able to search for it with the iPhone search feature (home screen-far left).

Is it a Gmail account with archiving 'on' instead of deleting (trashcan) : http://support.apple.com/kb/HT4207 ?

I don't have a Gmail account, but there is info on archiving here.

You, they have been moved into a an 'all mail' folder, which I'm not sure shows in the Mail app or not (I don't have a Gmail account) - if you can't see that folder in Mail then you should be able to via a browser. There is info about Gmail archiving on the second link that I gave.

From the front screen in mail, with the gmail account, you can select edit, pick the messages, the either touch the down arrow box you refer to, which moves to the archive, basically a storage place for all messages, or you can touch the file icon, which will bring up your files, such as Save, work, or whatever files you have set up. Also there is the trash can.