I don't know what Ballistol is. Never tried mineral spirits.
Acetone has worked in the past for me to pull out the dirt and oil.
I wear latex gloves, use the rough shop cotton clothes, and do the work in the garage only.
Rags only if you are going to be just applying a new linseed oil finish.
After having used steel wool before, I would vote no on it again unless you intended to sand the stock prior to the new finish going on.