I cherish my cat.

Good!

He despise him even punish him.

{He} is the third person singular, so add an -s to {despise} and an -es to {punish}.
Also please use {and} to connect two verbs.

He despises him.
He punishes him.

My dog is crazy he runs around In home.

Please use {and} to connect two simple sentences. Also, we say {at home}.

My dog is crazy and he runs around at home.

My friend admire jets.

{My friend} is the third person singular, so add an -s to {admire}.

My friend admires jets.

We crawl down the table.

You crawl down a hole.
But for a table, we say:

We crawl (below) the table

She is cherish to her new toy.

{Cherish} is a verb. We always say {cherish sth.}.

She cherishes her new toy.

He feel despise to his new teacher.

{Despise} is a verb. We always say {despise sth./sb.}.

He despises his new teacher.

I giggle because of the stink soup.

Please follow subj-verb-obj sentence structure.
Don’t use any conjunctions.

I giggle at the stinky soup

I had a friend his name is Peace.

Make sure you use {and} to connect two simple sentences.

I had a friend and his name is Peace.

he will shouts when he is angry.

{Will} should be followed by an original form of a verb.
And the first letter of a sentence should be capitalized.

He will shout when he is angry.