Can I use road salt on a shingled roof without causing damage?

Why would you want to do that? And no, I would not suggest it, it damages wood so I can be pretty sure that it would damage the roofing shingles.

Is there any roof friendly salt you would recommend?

No, I'm sorry, not that I know of, I'm not sure why you would need salt for a roof, though?

Sounds like you have ice dams on your roof and you need to get the back up to drain. I'd recpmmend Calcium chloride, your down spouts dump to the ground and the calcium will damage your trees and grass less. The salts will not damage your roof if they are able to break the dam and drain.

I would use Calcium chloride too I have used it in the past on my roof with all this snow it works really well and no salt stains either! I would first roof rake what you can then use Calcium chloride, you find it at your local poolcenter cheap!

Good luck