Is it permissible for a woman to work as a university professor teaching Western civilization and literature with the aim of demonstrating the merit of Islamic civilization and its influence, and raising students' awareness, while adhering to the شرعي حجاب (legal Islamic veil)? And is this work considered impermissible (haram)?

The default for a woman is to stay in her home, but she is permitted to go out for a need or a legitimate benefit, provided she adheres to the Sharia's guidelines.

If a woman's teaching is exclusively for female students, it is permissible, provided she adheres to the Sharia's guidelines.

However, if the teaching is for male students or a mixed group, it is not permissible due to the temptation (fitna) and unlawful mixing (ikhtilat) involved. This is an actual or highly probable corruption, while the hoped-for benefit is either illusory or merely presumptive. Averting corruption takes precedence over obtaining presumptive benefits.