It depends on the Application Support Center or Field Office. Some accept expired passports as a form of ID and some do not. You should take alternate forms of ID if you can such as school ID, birth certificate or national ID card if you have it.
You should consult with an immigration attorney with all the facts. It appears she may have the 10 year bar but I'm unclear as to why. Was there any misrepresentation issue? Had she been in the US before and overstayed her authorized period? Without more info, it's difficult to advise you.
I recommend that you wait for your EAD based on your adjustment of status through marriage to a US citizen. Since you did not find employment within the 90 days, your school may have reported it to SEVIS. Better safe than sorry.
If you are in current H-1B status and working, you should not change your status back to H4. Spouses of H-1B are not authorized to work.
Your issues seem more family law related for the termination of the marriage and child support. You should contact a family law attorney.
You do not have to pay the filing fee for your first I-765 under an asylum application. You must answer all questions truthfully. For the I-765 based on your asylum application, you will put the A# assigned to you.
I'm not sure about eligibility for a green card but have you already applied for DACA?
I recommend that you should see an experienced immigration attorney. If you do not have the proper documentation, you may not be able to successfully change your status.
If you have an approved I-140, it may not be necessary for new employer to file a new PERM if the job is the same or similar occupational classification. You should consult with an attorney to evaluate your case fully so that you do not make any missteps.
You should consult with an attorney to see if it's possible to administratively close your case based on PD (prosecutorial discretion) if your case is not strong enough to win and get permanent residence. If you get PD, you will still be able to renew your work permit every year.