I operated alot on gravel strips the 40 years I lived and flew in Alaska. If you do much of it, you will ding your prop and tailfeathers. You can minimize the damage but it will happen.
1. Bigger tires are better. I hope you have the tundra gear.
2. Do not do a run up on gravel. Ever! If you really must, do it at a fast taxi.
3. try not to park where you have to use alot of power to get moving. Manual labor (pushing) saves props.
4. Apply power slowly, no big bursts of power to get moving.
5. Stick always in your gut when you taxi.
6. Use flaps on take off. You want off the ground soonest.
7. Use all your flaps for landing. You want to touch down slowest.
8. No heavy braking on touch down. You can easily throw rocks ahead into the prop.
9. You want to be into the wind as much as possible. The prop will kick up the small stuff and if the wind is on the tail it will blow forward into the prop.
10. I've never seen a horizontal stabilizer abraision boot on a CT but if you can put one one you should do it. Get FD approval first.
11. Shouldn't hurt your oil cooler but you will see dings in the paint and prop. Will probably put some dings in the belly too as rocks are thrown. You will probably see wheel pant damage over time. Most bush planes do not have wheel pants.
12. If you have to turn around on gravel, as in you taxi to the end of the take off area then turn around into the wind, keep it moving. If you can't do that, get out and turn it around.
The last airplane I had in AK was a C-206. I flew it off gravel for 18 years and never did more than minor dings to the prop following the above advice. Just before I sold it I had the prop overhauled and the original blades were still good top go. I did see alot of damage to rental planes but most owners, having to pay the bills, soon learn what not to do.
Good luck.
John Horn CFII
2007 CTSW
Light Sport Repairman - Maintenance Rated
Rotax Service, Maint, & Heavy Maint. Certified
Independence Airpark, Oregon