Can generational curses or generational/family spiritual oppression affect Christians, even if they themselves have not intentionally opened those doors?
Yes, the oppression of family members can and does bother the Christian we hurt when our family is hurting. James chapter 1. But the devil will attack us but that is his nature has nothing to do with the generational curse. Job was attacked Jesus said the devil was a murderer from the beginning.
If so, what can/should the Christians do to free themselves from the influence of generational curses/spiritual oppression? (behaviors, prayers, etc.)
Christians personally do not have generational Curse because God said in His word HE would not hold it to those who are righteous the past sins of their fathers. Jesus comes and makes all things new.
For A Christian to have a generational curse means Jesus' power of the Cross is not absolute. That is unbiblical. "IT is finished " John 19:30. the payment of all sin was completed. All past, present, and future sin to those who place tier faith in the Lord Jesus CHrist.