Colleges in Chennai

Ofcourse yes, you can teach as you are a Masters and even having a doctorate degree.

You can check with the institution you are looking for and then find out whether they opt for the same or not?

As per the eligibility criteria, you are not eligible. The eligibility criteria of LLB (Hons) 3 years course is given below.
Eligibility : A candidate should have passed any Degree (UG) and should have secured not less than 60% of
the total marks in the aggregate. The candidates belonging to Scheduled Caste / Scheduled Tribe communities should have secured not less than 55% of the total marks in the aggregate of all subjects.
Because you have not completed you engineering degree course, hence not eligible.

What do you need an exemption in? There is nothing as such special exemption. Yes, you may get a preference over other candidates which is true but apart from that no preference.