First of all: in the first 1-2 weeks of 2019 we're planning to release version 2.15 of MistServer, which will include a significantly "pimped" meta-player.
Regardless, I can still comment/help with the other issues you posted. After all, we don't force any particular player, they _should_ all "work":
The HTTPS/HTTP issue is a browser-related thing (not JW-specific). Browsers will refuse to open HTTP resources from a HTTPS-requested page. You have several options to resolve this issue:
A) Serve everything without HTTPS (not ideal, but it does work).
C) Wait for the upcoming 3.0 release (planned Q1 2019), which will include HTTPS support in the free edition. Or buy the Pro edition if you want this _now_, of course. 😉
The error you're getting with the .ts link is understandable. This is a raw MPEG2 transport stream over HTTP. I'm not aware of any web-based players that know how to play that format (though practically every non-web-based player does).
Live MP4 and DASH outputs are probably what you're looking for in this case, which are both only supported in the Pro version for now (and no, these aren't being added as outputs in 3.0 either - sorry! 🙁)
WebM works very well when using the codecs VP8+Opus, but for most other codecs it only works in Chrome-based browsers. It's indeed intended as a method for very specific edge-cases.
Hopefully that gets you a bit further! Oh, and happy new year. 🎆