Your vehicle is being taken away by a tow truck! -- **Correct sentence~ ^^ 

She was just ticketed because she illegally parked in a handicapped parking zone. -- **Correct sentence~ ^^ 

He drove drunk, so his licence was canceled.

Better say: He drove drunk so his license was revoked. 

You need a valid permit if you pass.  -- **Correct sentence~ ^^ 

My girlfriend couldn't notice that I got my hair cut. 

Better say: My girlfriend didn't notice that I had my hair cut. 

I heard he played the piano, it was terrific.  -- **Correct sentence~ ^^ 

My friend can't open an account because she is a person with bad credit.

Better say: My friend can't open an account because she has a bad credit record. 

I sent him my photo ID, but he is not a police. 

Better say: I sent to him my ID, but he is not a police. 

A stranger pulled money out through stolen my card.

Better say: A stranger used my stolen card to withdraw money.

The campany required me an account for paycheck. -- **Correct sentence~ ^^ 

If you want to gain a profit, you need to make a saving account.

Better say: If you want to gain profit, you need to open a savings account.