Is there any way to tell if my Rod Thompson C14 carp rods are safe to use with braid--other than waiting to see if grooves eventually appear in the guides? What about the budget-priced Black Phantom rods that BCT, Resistance, and others sell?

I dont know. I think that if long as it gets grooves, just replace the tip with a different one like a fly rod tip top that wont wear down

Is it mainly the tip that gets grooved, or do all the guides get grooves from braid?

All the guides can be grooved but the tip is usually the main one.
If you like the rod a lot you could buy a set of tougher eyes and have them re-wrapped.

look at a the underside of one of the guide legs dose it say SIC

Are the SIC guides the only ones that stand up to braid?

The old fuyi guides with the green lining are good to but they are more heavy but if you dont have to cast far you can take smaller ones