Best video file for websites?

I use mp4 files and they load smooth. if it's larger than 500MB I'll run it through Handbrake first to make the size smaller while keeping the resolution the same

Depends on how much resources and bandwidth you want to use. I use embedded video sources and store and use their streaming resources.

I keep everything standard. I have some sites with over 100,000 hd, 1080p, and 4k videos. I use .mp4 with h.264 and CRF 23 in ffmpeg. I use 24 bit / 192 kHz for audio. For small files I would just keep it at CRF 25, 16 bit / 44.1 kHz. I never change the -r in ffmpeg because it messes up the frame rate.