When you open a new file and watch the Princess Peach getting captured cutscene Mario is already facing to the right on the world map (after the cutscene is over) But when you select a file where the cutscene is already watched Mario is facing down. (On the world map) So, wouldn't watching the cutscene be a small timesave? (Although watching the cutscene every time you fail would be annoying but a timesave in a speedrun is a timesave)

It takes the same amount of time to move from the starting point to 1-1 no matter which direction Mario's facing.

If Mario is facing down, he has to turn to get to level 1. If Mario is facing right, he does not have to turn which results saving time.

This is not the case. In this scenario, you need to hold any direction on the D-Pad for at least 2 frames to move. Mario facing down has nothing to do with how fast you can get to 1-1.

ohh ok