Correction:  Student's sentence   Corrected sentence        Grammar explanation       Better sentence            

I like animals.  Among them, I like cats the most.

This sentence is correct. 

Because when petting cats, I like softness, and when cats see people, they say 'Meow!'. I think the cats 'Meow' sounds like "A person who makes me look cute has arrived~!!!" 

When petting cats, I like the softness, and when cats see people, they say 'Meow!'. I think the cat's 'Meow' sounds like "A person who makes me look cute has arrived~!!!" 

And when the cats feel good, they are sounds 'Purr~Purr~Purr~' the reason I like that sounds because it is makes me psychologically stable.

And when the cats feel good, they are sounds 'Purr~Purr~Purr~' the reason I like that sound  it is makes me psychologically stable.

I was very happy when I had a cats. But I haven't cats now....

I was very happy when I had a cat. But I don't have cats now.

a is always followed by a singular noun - a cat

What makes me happy when I have a cats is that when I sleep, the cats play activity,  sleep well, and eat well, and the way the cats bake bread on top of me and my sister's belly or back! That's was very cute...(Cat bread is because cats like fold their arms and inside the body, which resembles baking bread!)

What makes me happy when I have cats is that when I sleep, the cats play activities,  sleep well, and eat well, and the way the cats sleep on top of me and my sister's belly or back! That was very cute...(Cat bread is because cats like to fold their arms and inside the body, which resembles baking bread!)

Cats baking bread has no equivalent word in English. You can maybe say cats make their body look like a brrad when they sleep.

There are cats in the apartment we live in, and they are so cute.

This sentence is correct. 

Good use of some words:

psychologically stable