CAT

It depends on the university you choose some may consider only Diploma GPA and some university also includes past GPA to obtain total result after completing Diploma (This is only applicable for those who has joined directly in 2nd year).

If 3.83 is the CGPA out of 4, it is good score in CAT application, so you have 98.9%.

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.