See clip from Attorney Rajiv S. Khanna's conference call video that addresses this question.
1. You can apply while your status is pending. but in my opinion an EAD can be applied on or after May 26th even if change of status to H-1 is pending.
2. Yes there is no law that stops you from applying a green card. you can start your green card , continue your green card or even finalize your green card while you are on your H-4.