A little assistance on the egg/secret boss? No spoilers, please!!!

I'm at the (I presume) final area, and so far the fast travel doors haven't unlocked yet. I have 9 pink coins, I assume the MysteryKey will help with the secret boss.

I have my suspicions on the egg room: the left door behind the first diner counter??? Idk, it may be too difficult to understand yet like chapter 3. I haven't looked around, to avoid spoilers.

Idk, I guess I'm asking for directions on these, but nothing too revealing. If anyone can help, it'd be much obliged! I gotta go to bed or smth. *See you in the funny stages!*