The inquiry is a bit iffy, as it depends upon whether the pull was made before or after they had accepted payment.
If they accepted payment as satisfaction of the debt, that terminated collection of any debt, and with it, their permissible purpose.
Inquiries are almost impossible to dispute even under the best of conditions, and I would not recommend disputing under your conditions.
I would call them and ask for their informal deletion based on the cirecumstances.
As for reporting after payment, payment of the debt does not negate the fact that they had legitimate collection authority, which they can still report.
However, they should have concurrently included in their reporting that the collection is both closed and has a $0 balance under their collection.
Kinda silly, as it no longer serves a business purpose on their part, but not improper.
They could argue that it serves the interest of the credit reporting system by accurately recoriding the fact that debt collector activity was a part of the history of the debt.
