CAT

ST category for getting IIT Bombay according to your category: The CAT score according to VARC is 50 and DILR score is 50

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

Yes you can expect a call from IIM Ahmedabad because your percentile is too good so I am assuring that you will get at a call with from IIM Ahembdad. Your cat score is excellent but you have to pass the personal interview also to have admission in IIM ahembdad.

337 is a very less score if you have the EWS Certificate or a state quota where the cutoff is less only there is a chance to get a college. Or you have to pursue for a private college. 337 is not decent score and chances are less to get colleges at this score but you should try in some private colleges where number of seats are good.

Absolutely Yes! You can get admission to a prestigious MBA college without cracking the CAT Entrance examination. There are many reputed Business Schools in India that offer direct admission in MBA and PGDM courses without CAT/MAT/XAT/CMAT entrance examinations.