Can my C-corp pay my country club dues?

You are in need of a good CPA for this and no doubt other question. Sure, the corporation can pay these expenses. All you have to do is write a check. Whether the car or any other expenses are legitimate, deductible expenses (which presumably is your real question) is a different issue. Are your dues legitimate business expenses that are deductible? Highly unlikely. Get a CPA - every business needs a good one.

I'm assuming that your question is really whether the country club dues would be tax deductible to the C Corp. The answer to that is likely no, unless the membership is used purely for business purposes. Even then, it would only be 50% tax deductible as meals & entertainment.

Dues are never deductible on a tax return, regardless of how much business entertaining is done. The actual expenses of the entertaining might themselves be deductible. Your business partner is only asking for trouble by considering this route. Yes, it would be taxable income to each of you, but it also might raise eyebrows in an audit.