Does damaging the brachial plexus mean permanent damage or will it help eventually?

Depends on injury. The cervical plexis can be stretched as part of the critical seconds during delivery.After the head is out, it is pulled forward & angled up or down to get the shoulders & rest of baby out.If the plexis responds with swelling around the nerves, transient loss of function(dys/wks) may be found.If the nerves are damaged by the stretch, nerves may be permanently damaged.The recovery potential varies.
Not necessarely. It's pretty common to have a transient brachial plexus injury during difficult vaginal delivery of the large infants. In most cases it's temporarily from the nerve stretching. Usually time will tell. Ot/pt intervention would be also helpful, ask your pediatrician for referral.