If someone has shared a file with me on Dropbox, can I save it on my computer, so that I have a copy in case they decide to delete it?
Of course, as long as you store it outside of the shared Dropbox. DO NOT MOVE the file, but copy the file.
If you move the file (simple drag & drop), the file will be removed from Dropbox on your machine and on the person who is sharing the file! In that case the person who shared the file will have LOST it!