Can a root canal or process of putting in a permanent crown lead to permanent facial nerve damage/paralysis?

The nerves to these different sites are physically separated, like separate branches of a tree. Work on/near the outer part of one branch does not affect the different branch (to your facial muscles).

As previously discussed in your prior question, dental procedures do not cause facial nerve problems.

No. Asked and answered. Asking it again and expecting a different answer is Einstein's definition of insanity.