MBA/PGDM

Yes, it is possible.
In fact, the Top colleges for MBA consist students, mainly with Engineering background.

IIFT is the best and well known college in Kolkata. You should try in that College.

For MBA admission, you should need to crack the entrance exams and with bases of your percentile, you will be getting admission. Your graduation percentage should be equivalent or more than 50% . The fees structure of college is different.

There is no any hard and fast rule that you must have to gain work experience in finance in order to get advantage, however it is always very much beneficial too.
Banking sector
Insurance sector
Stock and share market
Accountant post etc. will be very good thing to add up in your CV.

Salary is no bar. It depends on the sector like private, corporate and government. IIHMR graduate are getting good package.

Jobs are available in many hospitals, clinic, nursing home, KPOs etc. It has a high demand in the market.

It is not clear, what you mean by "Without MAT score".
With MAT score, k g somaiya, NITs will be good choices.
Without MAT score, I am guessing you will take CMAT (because, you are not mentioning CAT), there are very small numbers of good colleges like K G Somaiya. But, cut off will be really high.