Are the Flex-6000 antenna ports center and shield conductors short to ground when the radio is OFF?

Are the Flex-6000 antenna ports center and shield conductors short to ground when the radio is OFF?

No, they do not.

Also, would it be beneficial to have a surge protector in-line with the antenna in my current setup?

Yes, always - add one now.  The probability of a direct strike is very low.  The probability of getting a strong and damaging EMP (electromagnetic pulse) from a near strike is much, much greater.  I live in the #2 lightning capital of the US (central NC) and in the past 20 years, the house across the street has been hit twice (and I am 35' higher in elevation) and a very near direct strike (300') along with numerous strikes within a 1/4 mile.  I have never sustained any damage to my radios.  I do use an antenna switch to ground the center pin and disconnect the radio and I swear by the lighting arrestors based on the ICE design (now made by Morgan Manufacturing and Array Solutions).  In addition to handing EMP events, they also drain off DC static.  The 300' near strike smoked one of the ICE arrestors but left the radio equipment unharmed.  It was the best radio insurance I have ever purchased.

However, you need to make absolutely certain that you have a low impedance and high voltage connection between the arrestor and an earth ground that is outside.  In the case of your antenna, if your coax shield is not earth grounded, the lightning arrestor is not going to be very effective. The re-directed energy must have an adequate path to ground to be effective.   I have mine in a NEMA enclosure mounted to a copper backplane and the ground rods are no more than a foot away.