MBA/PGDM

Yes, PGDM is offered.
Post Graduate program in Management 2016
Average salary offer-: Rs 10.77 lakhs per annum,Median salary offer-: Rs 10.50 lakhs per annum and Highest salary offer-: Rs 15.5 lakhs per annum
Post Graduate Diploma in Management
Average salary offer - Rs 8.86 lakhs per annum and Highest

CAT, CMAT and CET are acceptable for admissions. Check below for entire process and details with scheduling of GD/PI:
http://jbims.edu/mms-2017-19-admissions-announcement/

Yes, surely if you score well. Check colleges accepting PGCET exam in the given below link:
https://www.shiksha.com/mba/ranking/top-mba-colleges-bangalore-accepting-pgcet/2-0-0-278-9262

Placements are not good and not renowned college, so my advice is not to go ahead with this college, you can try out Symbiosis Pune for Supply Chain Management.

GMAT is not necessary for Distance mode MBA.
Symbiosis, Welingkar, IGNOU, NMIMS, Amity and SMU are the best colleges for distance education and well-designed curriculum.

Placements are pretty average and nothing is special in this college, it ranks below 50 in top colleges’ list.
Refer the link:
https://www.shiksha.com/getListingDetail/22679/institute/college-International-School-Of-Business-And-Media-Pune-Pune-India

Kj Somaiaya and MICA have full-fledged syllabus on sports management and celebrity marketing.
D. Y. Patil and IISM are best for sports management specialization.

You have good options of doing part-time MBA from Welingkar and KJ Somaiya as they offer best grooming opportunities. You may also try out NMIMS and Welingkar for distance MBA.

You are eligible to get into the college. It is not necessary that you need to apply with CAT percentile. You can apply with CMAT, MAT or any other MBA entrance exam.