Can I use a debit card linked to my checking account: If so, is there a credit check and/or additional fees?

I recommend use credit card like Visa card, American credit card or other credit card.

The have a debit card policy. I would call them.

You can only use a debit card with a return flight itinerary. If you don't have proof of a return flight you must use a major credit card that is NOT debit but also is in the primary renter's name. With Budget or Avis when using a debit card the system does an automatic soft credit check. The results are provided by Equifax but the location doesn't know what are the requirements to pass, Avis/Budget's system just gives a yes or a no. If you don't pass the credit check you must use a major credit card that is not debit or you will not be able to rent.

Use it as a credit card, instead.