Been trying to figure this out.
I'm sure most people on this board will disagree with me here, as generally the best advice for personal finance is to develop a strong allergy to debt. It's, at least, good that you are avoiding a cash advance.
If you have the credit for it, you should really just get a personal loan. The rates are better, the repayment terms are better, it's an all around better environment. I have done this through credit card banks (Discover and American Express are good in this regard) and also through a peer to peer lender.
Barring the much better solution of a personal loan, depending on your specific credit card bank there are balance transfer checks. Again, depending on your specific bank, you can write the balance transfer check to yourself in order to pay a vendor who may not accept credit card or third party checks. You need to call your banks, and have honest up front discussions with them about what you intend to do.
Working around the rules of your bank to concoct illegitimate transactions not only will contravene your agreement with the bank but likely anti-money-laundering laws. Contriving a fake paypal invoice from a friend who cuts a check back to you will work but is, as far as I'm aware, illegal in the US apart from also breaching your agreement with the bank.
Start by researching a personal loan. It's the best option. Next look in to balance transfer options from various banks. Don't self deal.