I'm sure you know this but just to be clear you could start working for the new company based on the filing of the H-1B petition alone and without premium proce

I'm sure you know this but just to be clear you could start working for the new company based on the filing of the H-1B petition alone and without premium processing, it could be many months before the petition is reviewed (and hopefully approved). But yes, you can receive stock as long as you are not receiving it in exchange for services rendered (at least not until you are authorized to work for the company). My expertise is limited to US immigration law so I can't comment on your tax question.