I want to marry a Mexican Citizen. What do I need to do for her to be able to live with me in the US?

After you marry file the I130. She will have to apply for the immigrant visa (greencard) at the USEmbassy before entering the US.

You need to file the I130 for your wife ASAP. If your baby is born in Mexico, you can register him/her at the US Embassy as a USC based on your USC.

I am a Japanese citizen used to work illegally in the U.S about a year and overstayed few days.

I agree, I don't think you have to worry about not being admitted.

I just received my permanent resident status in the U.S. I've been on H1B visa for the past 8 years.

No.

What should my fiance tell the border officer?

If they suspect she will stay and get married here, they probably won't admit her.

White passport for my husband.

Do you mean a reentry permit? You could apply for one if you want to live abroad with your husband for 6+months.

Can my mom visit usa on her visit visa while her green card application is going through administrative processing?

They can try. If it's clear they are only visiting and intend to leave the US to apply for the immigrant visa abroad they should be allowed entry....

H1B Quota -Cap Case

If that petition is still valid and the same employment terms apply, you can use it with an updated employment letter. However, you will need to...

Will she be banned, or is there still hope for the spousal route?

Make sure you gave a valid marriage certificate before you file the I130. I suppose a conversation between your attorney and the US embassy is...

Married to us citizen, green card application

Yes, its similar as a nonimmigrant interview, but much more in depth. Success rate is similar. Its really up to you and your preference in terms...