There aren't any major tricks in this level. The fastest way to move up stairs is to continuously jump, however you want to run for a few frames between each jump as the stairs give you a little bit of speed in the beginning, but then don't let you exceed that speed.
The boss fight should ideally take no more than 3 egg shots per hit. 1 and 2 eggshots per hit are possible with screen-scrolling strategies.
This trick is the most saught after and prestigious trick, since it saves over a minute in a run if done.
The trick uses a long series of perfect flutters and then bats to regain height with extended flutters to skip the autoscroller, it gains its difficulty in being able to perfect flutter consistentently and manipulating the bats which don't do the same thing everytime.
The current warpless and 100% world records use this trick.
You can find an image based step-by-step guide, for help on learning the skip, here.
Sluggy 1 Shots
Here are possible setups for Sluggy one-shot hits. We're trying to find a good consistent way that would be realistic to do in runs.
There is a 1-1 warp at the beginning of 5-4 that saves quite a bit of time. This video demonstrates the warp in real time and includes a description for a consistent setup (courtesy of Blkyoshi) for the warp. (skip to ~9:45)
There is also a harder strat developed by Poahr for the 5-4 warp. None of the Warps runs have strat yet do to it being incredibly difficult RTA. https://twitter.com/Poahr348722/status/861134043227668480