B.Sc Electronics

To be honest, the educational background doesn't really matters when you are going for MBA.
Because, even if you look at the syllabus of CAT, MAT,XAT etc., they all include aptitude and Verbal.
It has nothing to do with commerce.
So yes, you can definitely go for MBA.
You will have to decide which field you are interested in. There are number of options marketing, HR, finance, IT, operations, supply chain management etc.

Yes, you can opt for M.Sc. astrophysics after completing B.Sc. in physics, electronics and maths.

No, you are not eligible for GATE with B.Sc in electronics (3 years). If your B.Sc is of 4 year duration then you are eligible otherwise after M.Sc you are eligible. Visit the link: http://www.gate.iitr.ernet.in/?page_id=266

Your basic qualifications just provide background. The remaining things are to be added. It depends on your interest, solely. Moreover, training yourself goes on lifelong. You can pursue a management course. Work for a career and you will succeed.