“If I use two eggs to jump to the gem…”: With a precise jump you should be able to get the gem with a single egg. Try jumping as high as you can to the left wall then throw the egg to the right wall to not lose vertical momentum in the small ceiling, other than that I think is just about optimizing a bit better. Good luck!
I finally beat it and found all gems except for the "Thank you for playing" stage, but that one doesn't have a door or a gem, I think. Thanks for your help.
My final feedback is that there should be a bonus for beating every stage with a gem, like an extra message at the end. I had fun.
Edit: Also the code to skip levels doesn't work, it only changes my browser tab. I tested it on a desktop computer and a laptop to make sure it wasn't a keyboard problem.