Is it possible to that the piercing has caused one of my milk ducts to clog form scar tissue?

Yes. When a piercing heals open, it forms a scar. Those milk ducts below the piercing will be interrupted and won't reopen. In spite of this, most women with piercings can still nurse.
It could. Especially if it caused a low grade infection after piercing.
Yes. Milk ducts empty into the nipple so if you have a nipple piercing it is possible for the scar tissue from the piercing to obstruct a duct. This presents as a painful lump. If the piercing is recent, remove the piercing and use hot compresses several times a day to cut down swelling. The duct may re-open. If it doesn't improve quickly, see your doctor.