You can send downloaded music file via iMessage, but you cannot apply that as ringtone. Apple wants you to buy those overpriced ringtones from iTunes Store.
All kinds, as long as the download is provided as standard HTTP transfer, not some obscure or proprietary protocol. For example, downloading files using torrent is not an option.
I know there used to be a way to make an mp3 into a ringtone using garage band for ios. However garage band has been updated and I can't figure out how to use it