Thanks for the detailed description.
I had no problem in playing the flv video, the only problem was the latency but that’s for my slow connection.
I can’t play this mp4, not on my site, not on the tester here
Anyway, the main problem you have, is that the player is not shown on your page, and that’s happens, as you said, when you edit the video details after published it.
You are probably having a conflict with another component of your site, this could be the theme or another plugin.
What you have to do is change to a default theme, something like Twenty Sixteen, disable all plugins and then reactivate them one by one.
That could be useful to find out the responsible of your issue.
Last thing… what happen if you use a YouTube video? Have you got the same problems?