Does over-vigorous sniffing mean water reaching the mouth from inside the nose, as in the Hadith: "And sniff vigorously unless you are fasting"? And is my ablution invalid if I do not sniff vigorously? And is expelling water from the nose (istinshar) for removing dirt or just water?

Exaggerating istanshaq (sniffing water into the nose) means drawing water with the breath to the furthest part of the nose. What the questioner does by letting water reach his mouth is beyond exaggeration, but it does not invalidate the wudu. Not exaggerating istanshaq also does not invalidate wudu. The obligation in istanshaq is to draw water into the inner part of the nose.

As for istinthaar (expelling water from the nose), it means expelling water from the nose after it has entered it. This includes cleaning the inside of the nose from dirt and mucus, and it is considered mustahabb (recommended) by some.

The primary purpose of madhmadha (rinsing the mouth) and istanshaq (sniffing water into the nose) is purification and maintaining external cleanliness.