Here are my thoughts,
1. Not getting into specifics, logically speaking, as soon as you enter USA on H1B Visa, your visa status is tracked as work visa. Your employer has to run your payroll from the day you landed in America. You can take vacation, but make sure your employer counts this as paid vacation. Usually, you would need to apply for SSN in the first week you land in USA to do your payroll processing. Your Immigration attorney is your best source for this, check with him/her.
2. I do not think, if it really matters. But again, it is worth talking to your immigration attorney.