Degree

Yes, you may pursue M.A. in economics, etc.

You need to specify the type of course that you want to pursue. Since you have done PGDM, you can opt for a job in some industry and if you are looking for online courses then there are many.

Healthcare is a sunshine industry. The scope is very good especially in Canada, USA and Australia. But if you are pursuing this course here in India then it's difficult to get the job directly abroad. You can work here for couple of years and then can apply for foreign. Apart from the hospital, there is a huge scope in healthcare business and Health IT consulting space.

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.