Does inserting a finger into the nose to clean out dirt, sometimes pushing it further in, have the potential to reach the inner cavity and break one's fast?

Nasal discharge (mucus) does not break the fast, whether it is dry or not. There is no obligation to remove any of it that clings [to the nose]. If the mucus reaches the throat and the fasting person swallows it intentionally, there is a difference of opinion among scholars regarding it, but the preponderant view is that it does not invalidate the fast.