I would first go to IC De la Ville, which is a little tricky. The hotel is on a one-way-street on top of the Spanish Steps, next to Hassler. The Borghese Park starts only a few yards away. However, I have not dropped the car at Borghese, I dropped it at termini; Borghese is not open 24/7! Ask at IC De la Ville how to get to Hertz. Going to Rome City Center is easy, just follow the signs. Watch out, very few streets are reserved for driving only with a permit...only a few!
Try not to check-in at the main check-in. Next, right to this (in the next room) is a "VIP" desk for check-in.