Hi
your questions are for YouTube/Google rather than Xerte really. See
developers.google.com/youtube/iframe_api_reference
However to answer your questions this partly depends on whether the video you want to use is from your own channel or someone else's channel? YouTube used to allow prevention of related videos with a parameter like rel=0 but I think at the moment what that does is still shows related videos at the end but only from the same channel from where the video is uploaded.
However if you can get loop to work it never reaches that end screen so doesn't matter where the video is from. I did a quick bit of testing and my recommendation would be to use the media lesson page and then you can use a YouTube link with parameters like the following
www.youtube.com/watch?v=9B6Eyr1K0RA?&loop=1&autoplay=1&rel=0
You don't have to watch that through just scrub to near the end and then play to see the loop working.
This seems to work at this point in time but YouTube/Google have a habit of changing and/or deprecating features (as you can see in the above link) so you'll need to keep track of that.
HTH
Ron