This is a bit more complicated than one answer can provide.
First - HTML5 is not a video format, it is standard for webpages and elements within them. The video in an HTML5 Player (using the video tag) is still some other format, such as .mp4 or .webm, made using some other codec such as h.264 or VP8 / VP9.
Google is using starting to use pure HTML5 based players on YouTube, with the VP9 codec (COder-DECoder, to make .webm video files. Lots of media people called it HTML5 video, but technically its HTML5 players with VP9 video.
The video itself doesn't end in .html5 or anything like that.
Second - There is not a single solution that will be great for every situation. An example of that is what Google / YouTube is doing. That VP9 codec is very good, but unfortunately the latest versions of it are not available in many of the editing products or encoding products we use or have access too. Its harder for guys like you and I to use it, compared to .mp4 with h.264 (which is available in dozens of free or low cost products)
Note that there will always be surfers / members / visitors that have some old phone or desktop that is not upgradeable to lastest tech - and they will blame you for making your site unusable, blah, blah, blah.
Third - The tech is always moving. The .MP4 file created today using the h.264 codec is the most common, in another year that Google sponsored webm with the VP9 codec may dominate, but right as I type this many high quality guys are working on HEVC/H.265 codec that will have same quality levels, but half the file size, etc.
Its a moving target.
Forth - Its unclear to me from your post. Are you talking about videos to stream in your site or videos for subscribers to download and save on local computer?
The process is different, or at least some of the settings should be handled differently, between those options.
Heck, if I was starting a site today with streaming video in the members area I would use HLS (HTTP Live Streaming), which was developed by Apple to insure video worked good on iPhones.
Its adaptive streaming - meaning each video is encoded in different resolutions up to 2160p (4K) and broken into tiny segments of 9 seconds each. Then, every 9 seconds the player checks the bandwidth of your user to see what they have and sends them the best size for their situation. This way you are offering top quality to fans that have high bandwidth, but not causing lag for those on smaller bandwidth.
If they are home with high bandwidth they get that size. If they are on mobile device on cell network that get smaller files, but if that same mobile device picks up higher bandwidth wifi they get higher quality / higher resolution stream - even if the change happens in the middle of a video.
Its also a bit more secure in its natural state, and can be made more secure depending on who is hosting it. (
HLS Content Protection - Amazon Elastic Transcoder )
Cheers