Hello Vicky, this is David. I'm glad to see your writing about your lovely cat. I'd like to correct it. You can point out my mistakes, too. ^_^

I had always wanted a cat. (The past perfect tense will show your yearning for a long time)  

One day my mother brought me a small kitten from her friend. (also you can say ''a friend of her'') 

Then (I think it better to use ''at that time'' make it clear) it (If you know its gender you should use ''he'') was a small ball of wool fur/hair (''wool'' refers to sheep, but ''fur'' or ''hair'' can be used with animals in general)  with blue eyes, who (you see? You used ''who'' to indicate the dog's gender, so you should use ''he'' rather than ''it'') was placed on the palm.

For a long time I couldn't think of his name. For two days I just called him "cat". But later I with my friend decided (or ''I decided with my friend that...'') he fit the name Stepash. (or ''Stepash fit him well'') 

She also has a cat and his name is Stepan.

The next three months, the kitten grew up and the colour of his eyes changed to green. Now he's a big cat with green eyes. He has red-and-white hair/fur.

I really like the combination of his red hair and green eyes. (very good!)

He's already almost a year old.

Every day I give him porridge and tidy up his toilet. (I do the same thing! But it's hard for me to clean up their poop, lol)

Once a week I comb him, especially his big, beautiful red tail.

In gratitude (or ''with gratitude''), he sleeps with me every night. (Wow, how kind you are! I don't dare to sleep with them!)

He priholdit (?) me in bed and sticks his wet nose on/onto my cheek. (Gosh! How intimate you are!)

He is very spoiled and loves to play. (also you can say ''loves playing'')

Sometimes I scold him for it because he scratches me and the furniture.

In summer he sleeps almost the whole day.

I think it's because he's hot.

Sometimes I walk him to play in the street.

At first he is afraid and does not want to go, but then I cannot leave him at home.

He really likes the street.
He is very gentle and sweet.

I love him very much and I think he is the best.

Way to go, keep it up!

Faithfully yours,

David