I am a BE (ECE) student. Am I eligible to study MBA finance?

Yes. You are eligible to do MBA finance. You have to give mainly CAT for that and based on your score and percentile, you can apply to various colleges. Also, there are other entrance exams for MBA but main is CAT. Check out this link for more information: https://www.shiksha.com/mba-chp1 Good luck.

Yes, after completing graduation you can apply for any kind of PG course and technical+commercial knowledge will give you an upper hand over others.

All the Bachelor degree holders are eligible for all the general streams of MBA such as marketing, finance, HR, operations, etc.