If you’re in fullscreen, the only way to play and pause a video is by clicking it, because there is no play and pause button in fullscreen. Playing/pausing the video by clicking the screen does not register as a command so if you try to play/pause it plays half a second for you but no one else then stops and goes back to previous position.
Yes, you are right. Only Watch2Gether’s player controls will trigger a change that is synced to all other users in the room. This is required to keep the playback experience in-sync.