The jump patterns in the top video avoid all the red coins, saving a half second on the score screen. The jump at :16 is perhaps the trickiest and you may need to flutter a tiny bit to make it to the next platform. At :19 taking the bottom route avoids the flower, again saving a half second on the score screen, as well as avoiding the risk of a bonus game. Tongue the Zeus guy repeatedly as you run through him to avoid taking damage.
In room 3 try to face left when collecting the Baby Mario star to get a little boost to the right as Baby Mario. At 1:02 quick repress B near the peak of your jump (similar to a perfect flutter) to make it over the wall. If you miss it you can still take the bottom route and make it to the next star in time.
If you have good movement after collecting the second-to-last star, you can skip the final star so you don't have to wait for the Baby Mario timer to run out before entering the goal ring. Make sure not to collect the flower when fluttering to the end since there won't be an easy way to consistently avoid a bonus game. Fast stairs: