No, the player should not show play or pause while the video is loading. There are more than two states the player is representing with only two icons. Adding separate buttons would not help that.
Yes, it should, because I might want the video to play immediately after it buffers, or I might want it to just buffer and press play in a few minutes.
Having a play and a pause button which to push to indicate the current state of the video while it buffers is very useful.
Also, to the parent, a button should display its action, so a "play" button will play and a "pause" button will pause. Except when the interface is designed by a web designer.
"OK, it's displaying a pause icon - is that telling me it's paused, or that I should click it to pause?"