I have a question about with my Father DS-260. Can anyone please help?

I have a question before I answer: How is that the I 130 does not carry his name as is in his passport?

Can I leave the USA while I am waiting to get my green card?

If you have advance parole you could travel on it.

Can I continue to work if my H1B transfer got denied, but the latest extension in progress?

Yes if you continue working for the original employer you can continue working.

Would Be i eligible to receive NVC letter? My priority date is 10 july 2019.

I agree with the professor. Besides, by the time NVC turns around the priority could retrogress.

You should follow the instructions USCIS provides on that page. You should have asked for it at the interview.

My us citizen husband is filing for my kids from previous marriage. They are not in the USA. What forms do we need to fill out?

I 130 for each child he is sponsoring.

Can I fly with k1 visa to other States?

Sure you can fly with your valid national passport to other states.

How to get a legal form to get a mother with kids to get out jail to take care of her kids?

The prison, is it a criminal sentence that she is serving or she is in immigration detention?

Can i apply for OPT(F1 - stem) while my AOS and EAD is on process?

As long as you are in valid F1 status and qualify for OPT you could.

Can i travel Outside the Country with a Green Card Extension Sticker. But my Green Card has been expired 3 Years ago?

If you have a valid extension you can travel.