Banker

I am in the US employed with a company with whom I have an H1 visa. I
have the opportunity to advise on a particular task for another
company. Would I be able to enter into an engagement contract with
that company as an individual, and could I receive compensation for
offering those service from that company?

My employer does not have any problems with my working on that
opportunity, but I do not wish to get them involved by having them
receive the payment on my behalf and later pay me.

How do people get on board seats, while on a H1? I wonder if they need
an H1 to get on the board - as they received cash and stock
compensation for board participation.

Any advise, or thoughts would be very helpful.

Thanks!