@jpm1982 wrote:
If someone shares a file with me, and then deletes it from their dropbox, does it still stay in my dropbox?
It depends how it's shared with you and the actions you take.
If they send you a shared link to a file or folder, you're just viewing the content in their account. To keep your own copy, you need to either save it to your own Dropbox or download a copy. If you don't do this and they delete the file, you lose access to it. If you do save it to Dropbox or download a copy, then you have your own copy of the data.
If they invite you to join a shared folder, then you both end up with a copy of the folder in your own Dropbox accounts, and anything one does within that shared folder will be synced to the other members. If you or they delete a file in the folder, it's deleted for both of you. Again, to make sure you have your own copy that can't be deleted, you'd have to copy the files to another location.