The answer to (a) (along with many other things) is to use Flexbox:

  .my-container {
    display: flex;
    justify-content: center;
    align-items: center;
  }

This should work for (b) as long as the height isn't in terms of a percentage:

  .my-video {
    --my-video-height: 120px;
    height: var(--my-video-height);
    width: calc(var(--my-video-height) * 16/9);
  }