Degree

Yes, they can join. The government opens the portals to apply in the entrance exams in mid-June and remember the portals remain open only for 5-6 days so be alert.

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.