Going to Home Country for H-1B stamping

It is part of policy of the Department of State.

Whether my short trip of one month outside USA will affect on my necessary 3 months residence in state before applying citizen?

No your short trip of one month generally should not affect your naturalization requirements.

I am a us citizen and want to help my husband obtain a green card, we just got marry a few months ago and am pregnant.

William Stevens provided you with a pretty complete answer.

There should be an address on the RFE for you to send the documentation. It is the Lee's Summit, MO address.

Is the status of a persons citizenship public knowledge?

No, if you are asking whether you can inquire about her legal status from government records.

Domicile

Should be USA if you are residing in the USA and planning to stay here.

No, you can only work with a valid EAD.

Can I get stop by the immigration at the airport if am making a international trip for unpaid taxes

You CAN get stopped but it is not likely unless there is a warrant for your arrest.

How long (Potentially) their green card cases might take?

https://travel.state.gov/content/visas/en/law-and-policy/bulletin.html : this site may give you some indication but an educated guess may say that...

Can he still apply for non immigrant visa under the Hranka Waiver before that date?

We would need more facts to better advise you.