Foam shot at the start: 
0 red coins: 
Top route vs bottom: 
The first screen is mostly straightforward platforming with one small trick at the end. Instead of using the boulder at the end of the room to get to the door, it's faster to take a shell with you, spit it out in the hallway leading up the door, and bounce off of that and up to the door. The optimal strategy for this involves spitting the shell upward instead of forward, as is done in the video.
In the second screen, it is slightly faster to skip the cloud and clear out obstacles using eggs but it is much safer to take the cloud, as getting hit even once probably means losing time compared to taking the cloud.