CAT

Any student from any degree in their undergraduate can apply for CAT exam.

The short answer is 'yes'. Even if you are in 3rd year of your college, you can still give CAT exam.

You are eligible to give the CAT exam but still cannot satisfy the 'minimum 50% in graduation' criteria.

Yes, you have to crack the CAT exam in order to make it into IIMs.

The simple answer is 'no'. If you wish to pursue M. B.A. in I. MS then you have to clear the CAT exam. GATE exam is for those who wants to do M.Tech.

Yes,12th marks are not important for MBA Admission. You get admission to MBA institute on the basis of CAT marks.

Following are the exam centre for CAT in Mumbai
Ahmednagar; Amravati; Aurangabad; Jalgaon; Kolhapur; Mumbai, Navi Mumbai; Nagpur; Nanded; Nashik; Pune; Raigad; Satara; Solapur; Thane; Ulhasnagar.

If you have scored in the range 70-90 percentile then you have more chances of getting to tier2 or tier 3 colleges.

I do not know about domicile cut-off, but following data are important as well for you, I guess:
XIMB Cutoff - Xavier Admission Test
Verbal Ability 60
Quantitative Aptitude 75
Data Interpretation 65
Overall Figures 90
XIMB Cutoff - Common Admission Test
Verbal Ability and Logical Reasoning 94
Quantitative Aptitude and Data Interpretation 84
Overall Figures 91