Does my studying and teaching become unlawful if I do something unlawful, such as going to university immodestly dressed? And do I sin if I ignore religious ideas about what is permissible and what is forbidden?

The student's commission of a sin, such as displaying her adornment (tabarruj) during her studies, does not invalidate the certificate she obtains or its use for employment. Ignoring thoughts is not disbelief (kufr). If these thoughts are whisperings (waswasah), then it is legislated to ignore them and turn away from them. As for someone who is not afflicted by waswasah, it is not permissible to proceed with an action until they know God's ruling concerning it. However, if they do not proceed with the action, then they are not obligated to know its ruling.