West Bengal

No, college offers film-making course through distance. And even if you come across such institutes, pursuing film-making course through distance mode is of no use. Film-making is all about the practical learning. So, I would suggest you to go for a regular course.

Yes, if you belong to an Army family.

Yes, all the courses like PGDM and PGPM in Kolkata and Gujarat are AICTE approved.
You can check the below mentioned link:
http://www.unitedworld.edu.in

Try to pursue it based on your interest but I would suggest you to purse chemical engineering but see to it if you are good in chemistry, go for it.
All the best.

Following are the good colleges to pursue Diploma in CSE:
1. Techno India University, Salt Lake City, Kolkata
2. AANM & VVRSR Polytechnic, Krishna District, Kolkata
3. Shree Ramkrishna Institute of Science & Technology, Sonarpur, Kolkata

Majority of colleges in India accept CAT, CMAT and GMAT which are must to attempt if you target all good B-schools. You can also give XAT, CET, MAT, SNAP and NMAT.

List of top 10 college in west Bengal for Lateral Entry:-
Jadavpur University-Faculty of Engineering, Kolkata
National Institute of Technology, Durgapur
Bengal Engineering and Science University
Govt. College of Engineering and Leather Technology
RCC Institute of Information Technology ( RCCIIT )
St. Thomas' College of Engineering & Technology
Aryabhatta Institute of Engineering and Management (AIEMD)
B.P.Poddar Institute of Management and Technology (BPPIMT)
BC Roy Engineering College (BCREC)

Amity, NDIM, FOSTIIMA, etc. are the best colleges for the same in Delhi. XIME, Amrita, Christ, Alliance, Vanguard, Acharya and MS Ramaiah are the best colleges in Bangalore. If you look beyond these 3 cities there are many better MAT accepting institutions.