An initial H-1? An extension? A visa stamp from a COS that took place in the US?
Without all the facts it is difficult to give good answers. So, please understand that these are guesses that shouldn't be relied on too much.
You should talk to the lawyer that did the H papers.
1. Yes, probably
3. There is no average ...each case is unique. I've heard of it taking 6-9 months sometimes
4.a. Probably not ... ESTA requires that you haven't been denied a visa
b. Probably not
c. Talk to the lawyer who prepared the H papers ... if you don't trust/like that lawyer ... hire a new one.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.
US immigration law does not cover work performed outside the US. If you are denied a visa, you are no longer eligible for ESTA.
Your employer needs to retain an experienced immigration lawyer, whether myself or one of my colleagues, to review all the facts, advise them, and handle the case.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.