Being asked to start teaching before being employed as a new faculty

Question

Suppose that a non-US person F is scheduled to join a US university as a teaching-focused faculty member. F's official starting is set to be January 4th and, at that university, classes start in the same week, and F will be teaching 2 courses in F's first term.

F has been inforems by HR hat F will need to apply for an SSN upon arrival in the US, and F will only be able to start working once F receives an SSN by mail, which may take 3-4 weeks. F's department head says this is normal, all foreign hires go through the same process. He also says F is expected to start teaching F's courses right away, even if F isn't officially working yet.

1. Is this indeed the normal process/expectation for foreign hires? Also, is it even legal for the university to ask new hires to start working without putting them on the payroll?
2. Does F risk violating the terms of an H1B visa by "working" without being employed/paid?

Entering the US before the term starts to apply for the SSN number earlier isn't really an option, as F can only legally enter at most 10 days before the starting date of F's visa.

(This was previously asked on academia.stackexchange, but it was suggested to cross-post this question here.)

Answer 1

Worth reading this document (pdf) from the US Social Security Administration (SSA). It describes the rules for Foreign Workers and Social Security Numbers and on the second page states:

Do I need to have my number before I start working?

We don’t require you to have a Social Security number before you start work. However, the Internal Revenue Service requires employers to use your Social Security number to report your wages.

While you wait for your Social Security number, your employer can use a letter from us stating you applied for a number, and your immigration documents can prove your authorization to work in the United States.

Obviously, you should confirm this approach with your employer.