Provided you are still in a valid nonimmigrant status when the extension is denied, yes, but not to H1B since as a general matter, that opportunity has gone for this year already unless your employer is CAP except.
Your working in your home country has nothing to do with US immigration law. I assume you are allowed to work in your country. If you are not, then it is not really "your" country and my answer is then - no.
No, you cannot travel on B1-B2 visas to work for the employer. those visas do not carry permission to work.
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You may work for your employer outside United States. You may also apply for L1b as long as you are in status when you apply. The H1B is no longer available this fiscal yearAsk a similar question