If the J2 has a valid EAD, then yes.

Can I remain in the country if I apply for an E2 investors visa while the Process is in progress?

yes you should remain in the US while change of status from b2 to e2 is pending.

Am I still on b1b2 visa (with what I've entered the US) given that I've applied for a change of status from b1b2 to f1

yes you are still in the status you entered with.

About green card sponsor

If you are asking if your cousin may file a family based petition for you, the answer is No, unfortunately.

Will I be out of status if my change of status is denied before the expiry of my i-94?

You wouldn't be OOS in that scenario

Filing I-485 before arrival of my father to USA

Adjustment of Status can be filed for those in the United States.

Traveling with wife having B1/B2 visa while being on F1 OPT

It should be ok for her to travel on b2.

Want to understand USCIS processing time classification category for H1B Amendment. Please help. Thanks

It's not clear what was filed on your behalf. USCIS processing times are provided by type of petition/form number, not by classification of...

I am trying to get a temporary resident visa to mexico and need someone to help me prepare the paper work.

You will probably need an immigration attorney in Mexico and also an attorney in the US to help with your return. For the US laws, you can use the...

CIS will first review merits of the expedite request. If approved, your EAD application will be processed on priority.