How do carbon fibre road bikes handle rough conditions?
Carbon fiber road bikes are generally perfectly capable of being ridden on less than perfect surfaces without sustaining damage.

Obviously you want to avoid large obstacles such as potholes, rocks, kerbs etc.   

28mm tires are definitely a good idea. Depending on how much poor quality surface you plan to ride on, you might want to consider looking for a 'mild' gravel bike that can take 35mm tires.