Yes, if you win your COR case:
1. You will receive a copy of the judge's order stating that you were granted COR and Adjustment of Status, not really good for anything but it's evidence that you're an LPR as soon as you walk out of the courtroom.
2. Immigration judge will notify USCIS that your EOIR-42B was granted and your card is to be printed.
If you apply for COR you first submit a copy of EOIR-42B and the fees to a Special USCIS Service Center in Texas. They will mail you a receipt and biometrics letters. After that you submit the entire packet to the Immigration Judge and he/she is the only actually adjudicating your application for COR/AOS.

EOIR-42B: "Application for Cancellation of Removal and Adjustment of Status for Certain Nonpermanent Residents"

Also Malignon is right, the main problem is that the quota is 4,000 and there's way more applicants than slots. In general it'll take 4 years to win the case during which you'll have to periodically prove that the reasons still exist. During this time you are eligible for an EAD so you're not utterly screwed. You'd file I-765 under category (c)(10) with the application fee to the Chicago Lockbox: USCIS Attn: FBAS P.O. Box 805887 Chicago, IL 60680-4120, with the EOIR-42B receipt as evidence.