embed quiz into html5 video - cue points in a rich customizable player

Flowplayer and Jwplayer are amazing players and you could do what you want, but in the free version you can't remove their branding. I try to stick to opensource projects you you don't worry about licenses.

I've done something similar as you need with Mediaelementjs. It supports all you need at the moment, and they will support the variable speed at some point on the future as announced in their website. However it has been like that for more than a year now since I started watching, so probably your "near future" is before their "coming soon".

I think your best free option will be Videojs. They also support all you need. For the variable speed there is this plugin.

Basically you set up a listener in the timeupdate event, and when it reaches the point that you need, you attach the quiz html to overlay the video area.

I have used flowplayer extensively. I'm just now starting to dive more into their new HTML5 version. The licensing for flowplayer is reasonable. It does have a plugin for doing subtitles and the cue point functionality works great. I also like that they have the Flash version that works well for fallback and I still use it for rtmp streaming.

I have built a video presenter with synchronized slides using the flash version which I'll be changing over to have an HTML5 version. I use it to produce speaker videos with Powerpoint presentations. Sometimes we have surveys (live) so I have also been planning to build in the option for a survey slide.

You can take a look at what I have built as an example here (it has some lose ends). Example synchronized video slide presentation (I have no affiliation with Flowplayer. WebVideoDesk, branded on the linked page, is a service I am planning to launch someday.)