M.Tech

If you want to do the hardcore coding then you have to go for M.Tech and if you want to be in a managerial role and want to manage your team who would be consisting of engineers then go for MBA in IT.

As per current market trends, job opportunities for BE+MBA graduates are higher in comparison to BE+M.Tech graduates. However, a gap in salary and job opportunities in these two combinations is decreasing steadily.

It is an excellent strategy to do M.Tech from any of the IITs by cracking GATE after B.Tech from a private university. It will help you get a good job and help in career growth as you will be known as an IITian.

Wisdom School of Management,Kochi
Insparc Technologies Pvt Ltd., Kochi.
Above are the institutions offering M.Tech in Kerala.

Yes, you can do ME from BITS.
The eligibility for admissions: A minimum aggregate of 60% in the qualifying degree examination.
You have to appear for an online test conducted by BITS throughout India. Before that, you have to apply for the examination online.
The test consists of Tier 1 English and maths, Tier 2 core engineering questions based on your branch.
The selection is done based on the performance in the test.
Check this link for more details:
http://bitsadmission.com/hdmain.aspx

BITS takes conducts the entrance test online for M.Tech. So, previous year papers are not available for practice, but you can take help from the samples papers available both offline and online for M.Tech entrance exam conducted by BITS. There are two papers, one includes English and maths second is your core branch paper.
So, you could practice from the above sources. Apart from it, if you are also preparing for GATE then the pattern is almost similar for respective branches which include both aptitude test type questions and core engineering questions.

There is no embargo on doing M.Tech or ME or MS after MBA. If you meet the eligibility criteria of doing M.Tech, you can do M.Tech after MBA also.

Yes, you may, but it must be a relevant subject and suffice to your career plans.