I wouldn’t recommend cash out refinance for such debt or to sell your property.

On initial funding (I'd imagine I'd need this to back whatever kind of loan as downpayment): could I initially fund this C corporation, marking this seed funding as debt or equity (maybe the former is ideal in the event I want to distribute a dividend/wage back to myself from the C corporation?

If I just got $50k from a property should I use some of it to pay down my personal debt or just use it all to reinvest into another property?

You can structure their investment as debt or as equity, ie a loan to the company (debt) or ownership in the company (equity).  

We have no student debt or outstanding loans except the rentals and our personal house.

Either it's paying down debt or earning more money somewhere else.  

For houses my expenses excluding debt or cap ex run about 36% self-managed,  50% with a PM.  

The tricky part occurs when the co-op is the debtor in a mortgage note secured by the property.

Other times a cash out refinance will not be suitable is if you don't want to take on additional debt or the new payment will take you cash flow negative.It seems you have a goal of how you plan to utilize the proceeds of the refinance, so that is a good reason to do the refinance, but you will also want to look at all the other factors to make you final decision.Best of luck!

Towards paying your mortgage, paying down other debt or savings?