In most cases positive trade equity is similar to using cash for a down payment. Cash money, however, may be required if there is a lien on the automobile that you are trading and the balance is greater than the appraised value of your trade in.
Applying a manufacturer's employee discount is not related to the loan, it lowers the price you will pay for a new vehicle. Since you are paying a lower price for the new car many lenders will find this more attractive which may improve the terms they offer.
Most lenders will accept a previous bankruptcy as long as it is discharged. For Chapter 7 bankruptcies there are lenders that will offer financing before the discharge but after the initial meeting of the creditors.
From our experience a cosigner may be considered for a buyer who does not comply with all the credit requirements of the considering lender.