What VISA is it you are on about? They are not extendible anyway - when they run out you need a new one.
If you are on about the visa WAIVER scheme (which it sounds like) that is not extendible under any circumstances. And it doesn't allow you to work.
I'd forget that job. You will need to apply for a work visa and during the application you will have to disclose your criminal record - you can't get away with lying as you have to produce a police document to show it or proving you have none. The fact that you have lied on previous visits to the US won't exactly help you either.