Generally, employers and employees are permitted to arrange their employment relationship in any way they choose, subject to certain legal and constitutional limits. That being said, even if you are not compensated for the use of your personal vehicle, you might still be able to gain an advantage of the situation by keeping track of your expenses and deducting it from your income taxes on Schedule A. An accountant or attorney will be able to help you with the calculations and specific rules regarding the deduction.
Generally, no. The Illinois unemployment scheme contemplates that employers will "pay" for unemployed persons (much like workers' compensation) as a general cost of doing business in the state. Thus, unemployment benefits do not come out of the state's general tax reserves. If the employer is not subject to unemployment insurance regulation (such as railroads, small employers, and agricultural employers, to name a few) then employees of them are not permitted to obtain unemployment benefits.
Christopher R. Minelli
Minelli Law Offices, LLC
NOTE: This answer is not intended to be legal advice and should not be construed in that way. This answer does not create an attorney-client relationship and no such relationship may be created absent a signed retainer agreement. The author is licensed in Illinois only, and his answer is for educational purposes alone.