MBA Entrance Exam

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

Top MBA colleges with budget fees-
RV institute of management -75000. Directorate of distance education 8230. Dr Ambedkar institute of technology 3700
Dayanand sagar college of management and information technology 63750. BMS College of engineering 25300. The Oxford college of engineering 70000. Xavier institute of business management studies 50900. government ramnarayan chelleram college of commerce and management -48,000. National institute of retail management 67000.

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

It is better to pursue a regular MBA degree rather than a distance degree. Because MBA is one of those degrees which require a lot of regular networking and working. You would not get all of those from a distance degree.

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.

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