Distance

You will have challenges in seeking admission in MBA through distance in government universities as you do not meet the eligibility criteria to seek admission in MBA.

You can do full-time Diploma in Engineering along with class 12th board in private mode. It will require extra effort and if you have the will to do both simultaneously without impacting your academic performance in both the courses, you may go ahead.

Bar Council of India does not recognise any distance learning degree and in fact there was long tussle between students and UGC in one side and bar council other side. Bar council has clearly mentioned and specified LLM through distance learning is not eligible for reaching position in bar council approved colleges.
Madurai Kamraj, Pune University and Annamalai offers LLM in distance mode.

You are advised to evaluate MBA with specialization in Operations ftom IGNOU by distance learning. It is one of the best MBA by distance learning in India. Study material of university is excellent.

As far as I know, no college is offering B.Des through distance.

You need a minimum of 50% marks in your graduation.

Bangalore University accepts the same.