MBA Entrance Exam

No, it is not at all necessary. Self study is all that matters. But take mock tets daily. Here are some good books which can be used for CAT preparation :
1.How to prepare for Data Interpretation for the CAT   TMH / Arun Sharma
2.Pearson Guide to Quantitative Aptitude and Data Intrepretation for CAT by Nishit Sinha
3.Quantum CAT by Sarvesh Verma
Know more about CAT on :
https://edudrona.com/common-admission-test-cat/#1464927769900-a0617f54-da37/?mref=tanu

You need to talk to the college once.
I'm not exactly aware of the fact

Here're the best colleges through MAT & CMAT:
MAT Colleges:
SCMS Cochin,
MS Ramaiah Bangalore,
Alliance Bangalore,ICFAIUPES Dehradun
CMAT Colleges:
IFIM Business School, Bangalore
Christ College, Bangalore,
UPES Dehradoon
Jaipuria Institute of Management, Lucknow
PSG Institute of Management, Coimbatore

PGPM stands for Post Graduate Programme in Management.Its a value added feature which is given along with the MBA degree.Its aim is to give maximum industrial exposure to the students.More than 60 percentile is required. IBMR Gurgaon is best among all,It is known for its placements and many certification courses.

you should focus on basic maths, English and on current affairs

There is no official document which states the fact. If you consider for the interview purposes, maybe some people might have a bias. But the interviewers of IIMs are experienced enough to not let these biases affect the decision making.

I will like to share IIM students tips. Go through this once. This is very helpful for me. And definitely helpful to you also.
http://www.mbaonemi.com/articles/entrance-exam/iim-students-share-tips-cracking-cat-exam/

I suggest you write a mock CAT to figure out your strengths and weak areas. Depending on your score you can figure out when to start preparing. Its possible to score well in CAT by preparing rigorously for 4-5 months if your basics are good.
Check this link to know more about CAT :
https://edudrona.com/common-admission-test-cat/#1464927769900-a0617f54-da37/?mref=tanu