Will forcep delivery damage my babys brain?

Maybe maybe no. Not necessarily and certainly not always, but it is a risk factor. It may be difficlt to differentiate from any damage related to the forceps delivery itself, or the condition or traumatic birth that necessitated this in the first place.
Rarely. In skilled hands, forceps delivery is safe and can reduce the chance of birth injury by assisting a difficult birth. By protecting the baby's head and speeding the birth process, there is a higher risk for birth injuries to the mother.