Every time the H-1 holder changes job, the H-4 holder does not need to re -apply for H-4. Your H-4 is good. You don't have to have different H-4 for the employer. Of course, every time the H-1 holders change their job they have apply for new H-1 but as long as you never violated status and H-1 holder has never violated status that H-4 continued to be good from job to job to job, you are good.
Your second question was what document need to file H-4 EAD?
Answer is we don't know yet. When the instructions for the new form I- 765 come out along with the form I-765 they will tell you exactly what you need to submit . I suspect probably marriage certificate along with evidence of couple of last pay stubs of your husband's, copy of H-1 receipt and a copy of your H-4 receipt or approval or visa stamped. That typically they should be looking for. But I don't have any basis for saying, this is just my guess.