If ex-wife agrees to the transfer of jurisdiction from AZ to TX, what are the procedures? Can we just draft an agreement letter?

You'd need to file a stipulation to give to notice to the Arizona court, and then deal with the courts in Texas. Your lawyer in Texas should be...

That depends upon the wording of the order.

I need to initiate a divorce and also sign over all rights to the mother of my child.

That's a prudent decision. Use the find a lawyer tab above to locate an attorney in the county in which your case could be heard.

Separated from mother of children took kids 350 miles away without permission and hasn't brought them back when she said she was

Yes. Seek enforcement in your local court - and get a local lawyer to represent you!

From date of marriage to date of filing of first papers in a divorce case.

If the plaintiff's affidavit and the other documents

No, that's not a problem.

USA Citizen. Divorced. will Marry a Foreign citizen ( Asian )

Your question isn't entirely clear - but in the immigration process you will need to submit all relevant and requested papers. Retain a local...

If my divorce is not final can i openly be in a relationship with another man?

Are there children involved?

My ex wife file for divorce by default I was never serve , and she got remarried, is there a way I can found the date she filed

Yes - go to the courthouse and look at the file, or if the case is in New York, look on the state court system e-courts online index.

Do I need a lawyer

Yes.