Can my nerve pain cause anxiety and depresion?

Nerve pain: Any long term chronic pain can do this. So the answer is yes.

Indirectly, yes: Nerve pain can come from many causes. Some might also cause physiological mood/anxiety disturbances. Without knowing the details of your pain - it's not surprising to hear of those problems too. If your energy and resources are being depleted coping with pain -irritability, worry and unhappiness are all quite likely - it makes sense. I'd avoid medicating them. Resolve the pain and see if they go.