My fiance lives in Russia, and I'm going to file fiance visa. I'm a citizen of USA. Will they check his background on intervie?

He will need to produce a police clearance for everywhere he lived since age 16.

In 07/01/2010 I become a permanent residence and it was my first day in USA. I got it by my wife. Now I am preparing N400 form

The dates you provided do not make sense, so meet with an immigration lawyer to discuss.

Apply for Citizenship

If you are married to a U.S. citizen you may be eligible to apply.

Can I get H4 Extension for my wife who is Paroled and my daughter who has used EAD for work?

Yes.

I received a notice of approval of advance parole I130. This was 10 days ago. There has not been any further activity.

I assume you mean I-131. You should get the travel document in the mail within a few days and if you do not, call USCIS.

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Will marriage at church affect my petition?

F4 is a sibling petition, not a parent's petition. Perhaps you mean your uncle or aunt filed for your mother when you were under 21. If your...

Can I enter USA with my 10year multiple entry b1b2 visa even after being mentioned as no cos and no eos on my visa ?

No, your visa is not void, however you should ask yourself why CBP put no cos no eos on that I-94.

My husband is now on hold with immigration shoul I hire a lawyer?

An immigration lawyer can only help you if and when ICE takes your husband into custody. Your husband's immigration rights will depend on his date...

Who helps the US citizen when it comes to divorcing a person with a green card that has a permanent residents?

A divorce lawyer can help you divorce your husband.

F-1 Visa will expire in 4 months. Currently in India, would like to come to States to do a certificate course.

If you have a new I-20, you may be able to do that.