I cherish my new computer.

Good!

My crazy cat jump and run in my house.

{My crazy cat} is the third person singular, so please add an s at the end of {jump} and {run}.

My crazy cat jumps and runs in my house.

They throw the eggs to Peop’s window.

You mean people?

They throw the eggs to people’s window.

A chicken cherish its eggs.

{A chicken} is the third person singular, so please add -es at the end of {cherish}.

A chicken cherishes its eggs.

They admire to the airplanes.

Admire something.

They admire the airplanes.

She despise to her old akelele.

What is akelele? Watch your handwriting!
Also, we say {despise something}.
{She} is the third person singular, so please add -s at the end of {despise}.

She despises her old friend.

My dog will shakes when he saw a rat.

{Will} should be followed by the original form of a verb.

My dog will shake when he saw a rat.

She need to had a rest because She bump her head when she jump.

Need to do something!
{She} is the third person singular, so please add -s at the end of {need} {bump} {jump).

She needs to have a rest because She bumps her head when she jumps.