The third background check in the same I-485 application (through marriage)

Yes, it is normal. Hopefully, it will be faster this time. However, processing time for Application to Register Permanent Residence or Adjust...

You never know. The best way to communicate with them is by email. Good luck!

Can my wife go for H4 stamping while my H1B transfer in progress ?

Your wife will need your approval notice, your marriage certificate, your most recent pay stub to get her H-4 if you are still in legal status.

How can I sponsor a visa for friend?

There is J-1 Au pair visa that your friend may try to obtain through qualifying program....

Can i apply for a visa while in the USA, and what visa should i apply for?

You can apply for adjustment of status in the US. You do not need to leave....

No, this is unreasonable to purchase a trip that far in advance. Good luck!

I tried to look up a case for a dissolution with minor children case and I didnt see a disposition of that case

You need to know in which jurisdiction the person was divorced.

Green card allows you to work for any employer. It does not look like your EB2 was fraud. Good luck!

How will having part-year or zero health insurance affect my Permanent Residency standing (and citizenship application later)?

So far I have not seen any question regarding health insurance in Citizenship context. I think you should be fine. Good luck!

Do we need to file an application for our 15 yr. old daughter for her US citizenship?

Your daughter became US. Citizen automatically. You can apply for her passport and certificate of Naturalization.