You can see the hatch chances of an egg failing on the hatchery main page, there's a link for it there called View Egg Hatch Chances. If it's fresh and fully incubated, they can't fail. If it's a spoiled egg, there's a 70% chance that it'll hatch/be mutant. For fresh eggs, there's a 2% chance that it'll be mutant.
Your Showroom will store pets and keep them from losing hunger and aging! You can find it at the link next to the nursery up top. https://www.rescreatu.com/gallery/ this link should also bring you to your showroom.
You can get one extra hatch from doing daily rewards, found here: https://www.rescreatu.com/dailyreward/
(it's under the activities page)
Other than that, you can buy hatches from other players if they are willing to sell.
I believe the Showroom doesn't have a max cap (I could be wrong), but each time you upgrade it'll cost more tu to store more pets. I'm not sure if there's a cap on profile pets.