Lots of great questions here…
If the videos are just little clips from cell phones then I keep them integrated with the main photo archive and do everything I do with digital photos: make sure dates are correct, rename files to reflect date, embed searchable metadata, etc.
If the videos are long (like converted VHS and super8) then I often organize those in a separate Video Archive, but using the same steps above. I just don’t think people expect to see a few photos and a 130-minute VHS tape.
I convert almost all video to MP4 format with high-quality H.264 video compression and AAC audio compression. I just did this process this week for a client who had 900+ videos in their collection. I do this to ensure their videos can be access across devices and into the future. I do all my video transcoding with Adobe Media Encoder. I rip DVDs with Handbrake.
To your last question, you should try those videos in some other apps. For example, does it open in the free VLC player?