1.I am carring my toy.

> I am carrying my toy.

~Be careful with your spelling. It might give another meaning or it will be misunderstood by the reader.

2.My mom buying many things.

> My mom bought many things.

> My mom will buy many things.

> My mom is buying many things.

~ Your sentence in no. 2 is not clear but here are your possible answers.

3.Dad is reading the very big book.

> Dad is reading a very big book.

~ We use the article "a" for general. It was not mentioned what book did he read.

Ex. Dad is reading the book Mind Games.

4.They are waiting for playing soccer.

> They are waiting to play soccer.

~  Some verbs are followed by to plus(+) infinitive

* to play

* to talk

* to sleep 

5.My dog is eating likes squrrl and rabbit.

> Subject and verb agreement (When the Subject is Singular, the verb is also singular.)

6.He is chacking his homework and tasks,tast.

> He is checking his homework, tasks and test.

~Be careful with your spelling and add more spaces to your sentence.

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Thank you for doing your homework, Nicky! Please take note of your corrections and study well.

~Take care always and teacher Cassie loves you. ^_^

Have a Happy weekend.