Once a file has been downloaded, the recipient can do anything s/he wants to with it.
There is nothing in Adobe Send that allows the sender to set an expiration date on the file.
There isn't a concept of files expiring, so I'm not sure how to answer your last question. Once you delete a file from your acrobat.com account, we delete it from our servers.
We chose Adobe Send because it seemed to be preferable to both Dropbox and Hightail.
Adobe Send lacks some of the crucial features of Hightail, so we may keep looking for a new service.