Hey everyone, I had the same problem after deleting a photo I didn't like.
Did some research online and found one guy with the best advice.
I eventually was able to retrieve my video from cache files.
Read the instructions carefully, it worked for me with the "ChromeCache View application" option (had the rely on date/time search).
Here's the helpful link
http://www.fastcompany.com/3026018/fast-feed/facebook-celebrates-10th-birthday-with-nostalgic-lookback-tool-that-doesnt-really-
Hope it helps, Good luck. Myrz
PS: Doesn't help much with the sharing though, you could upload the video to facebook after saving it from cache or post the url link. But kind of takes away from the share style.