CMAT

For GMAT exams u can write other MBA exams in india but difficulty level varies.

GMAT and CAT syllabuses are different so you might face some difficulty in cracking CAT.

Correspondance courses for CMAT will vary as per the coaching institutes, some keeps 3 months, some 2 months. You can check respective coaching institutes for short term/capsule courses for CMAT. I suppose most of the coaching institutes would have started the programs. Some of the best coaching institutes you can look for -
IMs, TIME, Career Launcher.

Books for CAT preparation are good for CMAT preparation-
Language Comprehension
Verbal Ability and Reading Comprehension for CAT by Arun Sharma and Meenakshi Upadhyay-
Verbal Ability and Reading Comprehension for the CAT by Nishit K. Sinha-
Verbal Ability & Reading Comprehension; by Ajay Singh-
Some good books to build vocabulary -
Word Power Made Easy by Norman Lewis
Barron Pocket Guide to Vocabulary
Quantitative Techniques and Data Interpretation
How to prepare for Data Interpretation for CAT by Arun Sharma-
Quantitative Aptitude; by R. S. Aggarwal.

Colleges accepting CMAT score in Pune are - Institutes of Management and Entrepreneurship Bharathi Vidya Peeth, MIT, Viswakarma Institute, SIBM, National Insurance Academy, Institute of Business Management, Vaikunth Mehta National Institute of Cooperative Management, Suryadatta Institute of Management and Mass Communication. Colleges accepting CMAT scores in Indore-
Institute of Business Management and Research- IPS Academy, Jaipuria Institute of Management, Prestige Institute of Management and Research.

No, you cannot get into FMS-Delhi with CMAT score. FMS Delhi accepts only CAT score.

Prepare good and focus on reasoning and English because most of the people loose marks in English. Then parallel focus on other things. Maintain a proper timetable then you can achieve it easily.

Prestige Institute of Management and Research, PG Campus accepts the CAT, MAT and CMAT.
Good luck!

Yes, you can get admission in CMAT without CMAT. You have to give CAT, score 98+ percentile for general category, and then you have a decent chance of getting into FMS.