Indian Institute of Technology, IIT Bombay

Specify the question.

In IITs, admissions are done only through JEE Advanced
JEE Mains exam is considered for admission into NITs and IIITs.

Well for M.Sc Degree at IIT Bombay, you need to give IIT JAM exam. I have given GATE exam (not JAM) for M.Tech, but my guess is that the admission procedure would be similar. Get a rank within top 150, after which are shortlisted for screening test+interview (sometimes only interview). Only after clearing those interview you are admitted. For an IITian the GATE/JAM exam is exempted but you need to clear a minimum GPA limit for this. And yes, maths is compulsory.

https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
This is the college predictor link. Here, you will get that what college you will get.

To pursue B.Des from Industrial Design Centre - IIT Bombay (IDCIITB, one needs to have a qualifying mark in the UCEED (entrance test conducted by the institute).
For further details regarding this course please refer to the link provided below:
https://www.shiksha.com/design/fashion-design/course/b-des-industrial-design-centre-iit-bombay-idciitb-powai-mumbai-255094

It's not necessary actually. It's bifocal subject. You have other options as well like Electronics, Scooter Science, etc. But, most preferred option in CS as you can score more marks easily in it.

General Category:
749 (Opening)
2036 (Closing rank)
OBC Category:
383 (Opening)
1037 (Closing rank)
SC Category:
157 (Opening)
425 (Closing rank)
ST Category:
173 (Opening)
231 (Closing rank)

No, with 44% you are not even eligible for direct B.Tech admission.

Check out the link below for fee structure of IIT-Bombay:
https://www.shiksha.com/b-tech/course/b-tech-in-computer-science-and-engineering-indian-institute-of-technology-bombay-powai-Mumbai-109575