General Nonimmigrant Visa

I have been in the US for six years. First O-1B expired on Feb15, sent in application for the 2nd O-1B already in December. The itinerary wasn't sufficient, response due by April 26th, ready to send in the completed itinerary plus additional letters of intended employment. My household is in New York. I am still here. Am I illegal now? Is there a grace period?

I am on L-1A (been about 4 years) from company A and in closing stages of getting a very good offer from company B. Both A & B are global corporations. For company A, I oversee business across continents (Both North America & Latin America). Company B is very keen on having me on board.
1. What are all the possible options on visa front so that I can join company B?

2. What is the quickest option that can help me get on to company B?

3. If I want GC, can company B file for it when am still on payroll of company A - if so, can it be EB-1?

What kind of evaluation do I need for my non-U.S. education credentials?

Where can I obtain additional information about the export control regulations?

Will a petition be denied if an export control license is required but has not been obtained prior to filing the petition?

Regarding Part 6 of Form I-129, does USCIS require a copy of the export control license if one is necessary?

For a CBP port of entry, what is the procedure available to seek supervisory review of an officer’s refusal to admit a visitor due to the period of time he or she was previously present in the U.S.?