Will I be allowed to extend my stay with H1 denial; meanwhile am waiting on my H4 approval

The lawyer who filed your application is the best one to seek advice from.

Does my father qualify for 40 Quarter of Coverage under the Social Security Act?

No he does not qualify and will need a sponsor.

Will USCIS work on the coming Monday 01/18 since it's a holiday?

No it is a federal holiday and USCIS will be closed.

Is it possible to renew F-1 student visa with a notice of arrest on my record?

Yes you can apply but there will be difficulties. Hire a lawyer.

I wanna return to the US but I got deported to Mexico

If you were deported from the United States you will have to stay out for at least 10 years and then have a family member petition for you in the...

What advise and steps to take can i get on cancelling my petition because of my wanting a divorce?

Your husband is the petitioner and would have to file the withdrawal. If you do divorce it will end the process you are currently seeking and you...

Can you please let me know if these 4 days will be considered as illegal stay in US? If yes, what are the consequences?

Talk with the attorney who filed your application.

What is the best solution i have?

You should apply for your mother to become a resident of the US. If she enters now on the visitor visa she can stay and you could start the...

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Background check for applicant under family petition package

The background checks will take as long as it takes. There is nothing you can do except keep asking for a status update.

Can a caregiver qualify for a work visa?

Yes a caregiver can qualify for a work visa. The program that seems best for you is the J-1 Au Pair program.