When applying for an employment-based green card, would the application be accelerated if I had a Director/executive title?

Titles alone do not define the preference category for any given employment petition. Job requirements, such as education and experience, do.

Getting hired full-time under a J-1 visa. Is that normal?

J-1 visas are for "Exchange Visitors," most frequently used by agencies; not stand-alone employers, and can be very restrictive.

Can I work while waiting on an I-20?

You have violated your status. You would have to obtain a new visa from outside the US and be readmitted to regain lawful status.

Which process is faster to bring my brother to U.S: 1-I petition for him 2- I petition for my mother then she apply for him?

Those 'categories" you mention do not exist. Generally, a petition from a parent would be faster than a petition from a sibling.

Is it legally to add a note on the contract that the employer should start the GC process under an EB-2 after three months??

That might be a matter of labor/employment law, more so than immigration law.

What documents do I need to submit to prove I left on advance parole when renewing my DACA application?

The documents that you reference should be adequate to prove that left and reentered with advance parole.

How to change sponsor for non-immigrant visa?

Your new sponsor will need to file a brand new petition on your behalf.

We are moving to south africa. we are in the process of applying for temporary permits

You need to contact the South African Government. This form discusses US law.

Your cousin needs to consult with an immigration attorney fast.

221(g) means that the consul believes the person is not qualified for the visa and/or admissible to the US and will review the case further.