the revive pets option in my stable's isnt working, how do i revive my fainted pets?!
Also how do i feed my Sylestia's and care for hatchlings and eggs?

You can just give the pets an hour so so and they will auto heal themselves. (Or you can use a revival draught from your inventory page to speed things along.)
To feed pets go to your stable and in the top right there are several options. One is feed pets. You can click that and it will have a pop up that tells you the costs. You can confirm and feed them or back out of the pop up.
To interact with hatchlings and eggs you go to their profile and click on their picture. A pop up will appear and it will tell you what your pet needs and you will have several options. (Ex: Unnamed egg is cold: you should select 'warm egg') you can only interact with a pet once a day. If you interact with an egg correctly it will help the egg hatch faster, if you interact with a hatchling correctly it will get a small stat boost.

(Also if you would like I can offer you a higher stat pet to help you in battle and on missions to make gold.)