Is it bad to ride a horse on asphalt or hardish -slightly rocky- dirt?

Asphalt is okay to walk on. It's best not to trot or canter on it--too hard for the joints. Hard soil with some rocks can bruise the sole of the foot if you are moving faster than a walk if he steps on the rocks. If its only a few small rocks then trotting or cantering are probably okay. You should always make sure your horse is in condition for the pace you ask to move at and the length of time you ride at that pace. Tired horses are more likely to injure themselves. Good luck with your horse.