I'm glad you liked it, and that you found the easter egg :)
The wall mechanic is very reliable and precise once you get the hang of it. The problem is I don't do a good job of explaining in the game how it works.
When you jump, if you touch a wall on the upward part of the jump, your horizontal velocity becomes vertical and the cat runs up the wall. Nothing happens if you touch it while falling.
This is why, if you're right next to the wall and jump the cat drops back down quite quickly, because it has little to no horizontal velocity to convert into a wall run, and why the stairs are such tricky to get right. I will look into revising that specific behaviour soon.