NIMCET

Nope, you have to appear in NIMCET to get into MCA course after clearing the cut-off.

Yes, you can do MCA after B.Ed.

Yes, you can do MCA after B.EI.Ed.

Hi Atul Kumar
Kindly predict your college in this college predictor. JEE Main rank predictor is an online tool which predicts the candidates rank range and probable colleges based on the difficulty level of the examination, average marks scored in JEE Main and seats available in IITs, NITs, GFTIs, and other SFTIs. The rank predicted might not be accurate but it certainly does give an idea to the aspirant. Know your college admission chances based on your JEE Main Rank, Home State, Caste, Gender, etc. Here https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
Hope this helps!
Good Luck. !!

According to eligibility criteria, BBA student can give NIMCET exam. So, please participate in this exam.

Yes, NIMCET is a country level exam for admission into MCA course.
Good luck.

Some colleges will not have any problem if you studied maths in 12th or not for BCA course, but some colleges have the eligibility, criteria that student must have taken maths in class 12th. Therefore, it depends on the college you are applying for the course. Once you get admission in BCA you can then surely give NIMCET exam for MCA course in NITs.