Should the husband inform his wife's workplace – she who is seeking khulʿ after he discovered some of her suspicious behaviors – about her actions, given that he is certain of them, and should he accede to her request for khulʿ?

You should have chosen a righteous wife in compliance with the Prophet's command. However, since you are already married, you must advise her, enjoin her to do good, and forbid her from evil. If she responds, then that is good and excellent. Otherwise, divorcing her is a salvation, and if she requests khul', you should agree to it. It is not permissible to inform her workplace of her sins, for Allah is the Concealer and loves to conceal. However, if she will affect the students' behavior or her teaching quality, then it is obligatory to report her to prevent general harm.