Its hard to put simple but there is a certain pixel to full jump off of the elevator, then you release b in midair but continue to hold right. After that the walk for one frame then do a one frame jump then press left to clip in and then start running.

The backwards jump at the end you have to push jump on the frame you land after you bounce off the pipe in the warpzone.

Here's a video that might help.

Uhhhhh... B------ A B A L..............
Hahahaha idk

You should practice on the Pellsson Rom.
https://github.com/pellsson/smb/raw/master/smb-v5.6.ips

Yeah, pellsson makes it so much easier. If you have pellsson, here's a quick thing explaining it:
You want to go through the level perfectly like the video up there does. When you get to the area between the two places where there are platforms moving up and down, the subpixels in the 2nd to right value must be rotating D and 5. (If you want to know how to turn these on, pause, scroll down to where it says "show rule", click A, then unpause and you should notice different values in the upper left corners). I'd recommend making a save state on top of the row of bricks (dw about high of low way of doing it they are pretty much the same afaik) and maybe getting a bit of slow-mo because that helps sometimes (might be different for you though lol). Now, when you run off of the row of bricks, jump at frame 035 off of the platform. To know where this is, you can look at the thing in the middle of the top of the screen that says "FRAME" and if you jump right it will say 035. Now when you jump off, you want to do a full jump. Before you land, release B then jump almost immediately after landing (it might be a couple of frames later or it might not be, I don't think it matters though) and continue to hold right before hitting the pipe. Right before hitting it, tap left to hopefully pull you into the pipe. If there is anything that needs more explaining or if there is anything anyone thinks I got wrong, please tell me and I'll edit this!!! I hope it was helpful, I don't know what your skill level/how much you know so I tried to explain it in a pretty pare bones matter. GL !!!!!!