Degree

MBA is a Post graduate programme, so you first need to complete your graduation. If exploring BE/B.Tech is difficult, then you can explore BA through the correspondence route and complete it in 2 years' time. Just ensure that you clear BA with more than 50% marks. Post it, prepare for CAT & then get into MBA.

Yes, you can pursue your masters in psychology after Bachelors in engineering. It does not really matter, the background from which you come, but you need to have knowledge on psychology.

This is a PG degree in MBA.

Unfortunately, No.
You cannot directly enroll in the second year, because the university rules dictate that the student registration needs to be done in the final year, only unless you have studied in the same university for the same program just someplace else.
So, you have to enroll for a 3 year B.Sc program and do it from first year.
However, if you want, then you can also enroll for a PG (NOT M.Sc) course, which would be basically advance of the knowledge that you already have and as it is, experience and knowledge is what counts in this field, so that can also be a good option.

Some of top private BBA colleges in India are as follows:
Ness Wadia College of Commerce
Department of Management Sciences, Savitribai Phule Pune University
MIT SOM College
MIT WPU UNIVERSITY
CHRIST UNIVERSITY
Brihan Maharashtra College of Commerce (BMCC)
AIMS Institutes
ABBS - Acharya Bangalore B School
PES University PESU.
FOR DIRECT ADMISSION FEEL FREE TO CALL

You will have to clear intermediate first to pursue a graduation degree. So I suggest pursuing Intermediate and then any course in graduation preferably BBA.