Flowplayer and Jwplayer are awesome players and you can do what you want, but in the free version you cannot remove your branding. I am trying to stick with open source projects that you don't care about licenses about.
I did something similar as you need with Mediaelejs . It supports everything you need at the moment, and they will support variable speed at some point in the future, as announced on their website. However, more than a year has passed since I began to look, therefore, probably, your "near future" to their "soon".
I think your best free option would be Videojs. They also support everything you need. For variable speed there is this plugin .
Basically, you configure the listener in the timeupdate event, and when it reaches the point you need, you attach the html quiz to overlay the video area.
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