how to add animals in stable and how to make animals that we design added to stable?

For "How to make pets you design" you'll want to refer to our guide here on generator essences and how to use them.
IT covers the following points:
Post 1
Section 1: What is an Essence?
Section 2: Types of Essence
Section 3: How to get Essences
Post 2
Section 4: How to use an Essence in the Generator
Section 5: Non-Generatable Traits
Section 6: Non-Generator Essences
Post 3
Bonus Section, Section 7: Regenerable Pets
Post 4
Section 8: Generated Pets' Stats
Section 9: Visible Traits, Carried Traits, Hidden Traits


Not quite sure I'm getting what you mean by the first part of the question, but let's try:
You can obtain pets via catching them, making them in the generator, buying them, etc.

Pets hatched from eggs will automatically go into the stable (so long as you have free space) once they hit Adulthood at 27 days

When you're trying to catch a pet, if you succeed they'll go straight to the first open slot of your stables (you'll get a link in the log), though if you fail to catch the pet it might flee from the battle

Buying a pet, it'll either go to the Stable or Hatchery, depending on if it's an adult or not.

Generated pets go straight to the stable, I think?, but they're adolescent