Most video player clients and browsers will attempt to stream the video if they can.
For an mp4 video file hosted on a server, so long as the header is at the start and the server accepts range requests, this will mean the player downloads the video in chunks and starts playing as soon as it has enough to decide the first frames.
For more professional streaming services, they will generally use an adaptive bit rate streaming protocol like DASH or HLS (see this answer: https://stackoverflow.com/a/42365034/334402) and again the video will be streamed in chunks, or segments, and will start playing while it is streaming.
To answer your last question you need to be aware that the raw video is encoded (e.g. h.264, VP9 etc) and the video, audio, subtitle etc tracks stored in a video container (e.g. mp4, Web etc).
The most common format is probaly h.264 encoded and mp4 containers at this time.
The particular profile for h.264 can matter also depending on the device - baseline is probably the most supported profile at this time. You can find examples of media support for different devices online, e.g. for Android: https://developer.android.com/guide/topics/media/media-formats