What are the limits of interaction between males and females in co-educational schools, and is it permissible to shake hands with girls or sit with immodestly dressed women for the purpose of inviting them [to Islam]?

It is not permissible to shake hands with women, and there is no necessity for it, even if people oppose you. Their request for you to disobey the Creator is the true degradation. It has been narrated in the Hadith that whoever seeks Allah's pleasure through people's displeasure, Allah will spare him their burden. And whoever seeks people's pleasure through Allah's displeasure, Allah will entrust him to them.

As for sitting with women who are dressed yet naked for the purpose of da'wah (inviting to Islam), its harm is greater than its benefit. It is more appropriate for this to be done through righteous women, to protect you from temptation, as warding off corruption takes precedence over achieving benefit. They can be invited through means and methods that protect you from their evil and ensure safety from temptation.