Indian Institute of Technology, IIT Bombay

The IIT-B cutoff for B.Tech courses are varying according to specializations. The cutoff ranks of CSE goes very high. The last year cutoff ranks of some specializations are given below for an idea.
1. CSE (Closing rank 21)
2. Civil (Closing rank 661)
3. Electrical (Closing rank 100)
4. Aerospace (Closing rank 564)
5. Chemical (Closing rank 604)
6. Mechanical (Closing rank 250)
So from above you have to score at least 220 marks in JEE Advanced for getting admission here in Civil engineering branch.

For IIT BOMBAY you have to secure the top rank in the JEE ADVANCE exam.
From JEE MAINS you get the NIT's college not IIT'S.
I am sending you an authentic link from where you get all information of cutoffs for IIT BOMBAY.please check it out.
link-https://www.shiksha.com/university/iit-bombay-indian-institute-of-technology-mumbai-54212/cutoff

The eligibility criteria of M.Des in Mobility and Vehicle Design from Industrial Design Centre - IIT Bombay is given below.
Eligibility criteria : Bachelors Degree in Design/Engineering/Architecture/ Interior Design: 10+2+4 years (minimum four year Bachelors course).
Note: B.Plan. considered equivalent to B.Arch. Four year B.Sc (Engg) course after HSC considered equivalent to BE/B.Tech.
OR Professional Diploma in Design: 10+2+4 years (minimum four year Diploma program).
OR BFA Sculpture (4 year professional program with entry after 10+2).
So from above B.Des is not mandatory for the asked course.