Ph.D

Yes, definitely you can. But I suggest you not to change your domain. Since, you are not familiar with Masters level management terms, it can be difficult for you to recover so many things, in just a little time period. Better proceed in the same domain.

MMS is better for further studies like Ph.D & for government jobs also.
PGP by IIM'S also preferred.

Your approach and strategy to become an astronaut is excellent. You are further advised to learn flying and obtain CPL license. With your proposed educational qualifications and 1000h of flying you will become eligible to spy for astronaut job.

I don't think there is any such requirement in any sector.

If college will fund your stipend for your Ph.D, it would be a better option. If not, it will take a long time to get out of this course and to find jobs either in the education field or in any other field. No company will recruit Ph.D graduates directly at the initial level because of the fact that they need to pay a higher amount for the same job that would be done by a UG student. Please select carefully. Only if they provide stipend during your Ph.D then opt for the same.
All the very best.

VIT Vellore should be preferred over SRM since VIT has must better affiliation to the foreign colleges and you will be benefited much better for your foreign education.

No, it will not be advisable. Regular MBA demands your complete time and attention. You will be required to clock almost 10 hours at college and pursuing PhD along with it will not do justice to either of the courses.