It depends on the OS and Browser you are using, but usually all videos are saved under "Downloads" folder on Windows and Mac. You can also press CTRL+J in your Browser to view your download history.
That's something normal to happen, especially on browsers other than Chrome. To solve this issue, instead of left clicking the Download Video link, Right Click -> Save as... and choose the location you'd like to save the video to.
Yes, you can download Facebook Live videos but only after they finish streaming.
FREEFBDOWN doesn't store videos neither we keep copies of downloaded videos. All videos are hosted on Facebook's servers. Also, we don't keep track of download histories of our users, thus making using Free Facebook Video Downloader from freefbdown.com totally Anonymous.