Does my previous one time participation in dv lottery impede my USA-university sponsored j1 visa processing?

I disagree. I would indicate that you applied but fully intend to comply with the terms of the J-1

Divorced, Want to marry my father - sister's daughter. She is in Pakistan and last time we met was at my wedding. Im usa citizen

Most states do not recognize marriage between first cousins so that is the first thing to check out. The usual way to show a good faith...

I am in the process of bringing my mother from Peru. But can she bring her grand kids that she has legal custody of?

No unless they can get their own nonimmigrant visa or qualify for humanitarian parole.

Check with the British Embassy to see if nationals of your country need a visa with a U.S. green card.

If I marry a US citizen and apply for a green card will it affect my current L1B visa?

No the L-1B can continue until the green card is approved. You probably want to apply for the work and travel permit through the green card...

My husband already has his Employment Authorization Card, can he start working?

The EAD card is a List A document on the I-9 which he completes when he starts employment.

Please I need help as soon as possible.

You need to follow the instructions for the various forms on the USCIS website and the accompanying INA Act and regulations or better yet retain a...

Can my fiancée travel to the US on B1/2 while k1 is pending?

The border officer has discretion. Just make sure she doesn't lie. If they deny she should ask to withdraw her entry and await the K-1 visa.

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I am on Deferred Action for Childhood Arrivals (DACA) can i travel to puerto rico with out any immigration problems

You should have an attorney analyze your case before you do any travel or apply for an advance parole. I understand that Puerto Rico is a U.S....

If you are eligible for AC-21 portability or can wait for this then no; otherwise yes unless the permanent job offer can remain for the original...