Yes. Nerve injuries do recover if the ends of the severed nerve are sewn together by a hand surgeon. Beware that it is a very slow process. Nerves grow back 1 inch per month from the cut end out to the finger tips, so expect months before feeling may start to come back, though initial feeling may be more like the pins and needles of a nerve waking up before any quality sensation comes back.
Nerve damage recover. Yes. However, it tends to be quite slow and related to your overall health and age. Younger patients tend to demonstrate the best recovery with nerves growing an average of 1 mm a day or roughly 1 inch a month.