Yes you should indicate all children where indicated.

The overstay and unauthorized work is waived through marriage to a USC in good faith.

Humantarian reinstatment

There is no set time for humanitarian reinstatement. Contact USCIS directly to inquire about the status.

She should wait 5 years since GC issues or be prepared to establish that the first marriage was also in good faith.

If the pentitioner of the form I-30 lives in CA but the applicant lives in NY, where should the applicant apply in NY or CA ?

You would both indicate your physical and mailing addresses and their interview will likely be in NY.

No not if you are otherwise eligible to renew and indicate proper category for filing.

Immigration ?

Your friend should consult with an attorney to determine what, if any, immigration options are available for her.

Can I apply for petition for my mom while I'm unemployed?

Yes you may - assuming you are a USC your mother is otherwise eligible, and you have a cosignor available at appropriate stage of application.

If the error was on USCIS part, you do not need to pay the replacement fee.

Hi. I have been married to my US citizen spouse for 1.5 year. Now we are going to apply for my green card.

Yes you will receive the 10 year GC if case is approved after 2 years of marriage.