You will be able to apply for EAD 150 days after your application was filed and is pending. On the basis of EAD you'll be able to get a SS#. The EAD card, SS# and your passport should be enough to get a driver's license. Buying a car is never a problem. Hope you have a good attorney.
Please be aware that during the government shutdown, SSA is not issuing new or replacement SS cards. Obviously, do not wait to start the other paperwork that was suggested by Mr. Palant, but just be aware that your wait for a SS card may be longer than normal because of the shutdown.
You can try with your EAD. If denied, as my colleagues suggest .. you can get a SS#, after the US government re-opens for business.
PROFESSOR OF IMMIGRATION LAW for over 10 years -- This blog posting is offered for informational purposes only. It does not constitute an attorney-client relationship. Also, keep in mind that this is an INTERNET BLOG. You should not rely on anything you read here to make decisions which impact on your life. Meet with an attorney, via Skype, or in person, to obtain competent personal and professional guidance.