Wow ... these are a lot of questions ... demonstrating that this can be quite complicated.
You need to talk to the Company B immigration attorney. Normally there are no grace periods (ie: you shouldn't be w/o work from Jan 11 to Feb 11 ... much less March.)
I understand that some time is needed to move ... see if company B is willing to pay for moving expenses and if the lawyer agrees that this is part of the normal company employee benefit program.
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.