What is the ruling on looking at the face of a female student who has her hair uncovered when there is a need, and is that permissible? And does the need for education permit committing the forbidden act of looking at the face of a female student who has her hair uncovered?

A woman's hair is considered 'awrah (private parts that must be covered) by scholarly consensus, and it is impermissible to look at it. Looking at a non-mahram (definition) woman's hair is not analogous to the permissibility of looking at her face for educational purposes, as looking at the face is permissible out of necessity.

Given the prevalence of women's adornment (tabarruj) and the resulting difficulty and hardship from an absolute prohibition of studying, using public transportation, or frequenting markets, if you need to work and find no way to teach males or veiled women, it is permissible for you to teach unveiled women while lowering your gaze as much as possible and reminding them about the hijab (definition). If they do not adhere to it, they must be prevented from wearing provocative clothing.