Colleges in Bangalore

You should have given CAT exam because this time the admissions are being done only through CAT results.

Please use the college predictor tool by Shiksha. This will help you to predict the college according to your rank/percentile:
https://www.shiksha.com/college-predictor

That's a reasonably good score, with that score your son can quite easily secure admission in Christ University. Provided he qualifies any other requirements laid out by the University.

If a candidate scores good marks in entrance exam for BS conducted by IISER then a candidate can apply for this course.

Yes, you will get admission in 2nd year. of B.Tech as lateral entry. For lateral entry, normally you have to appear for Lateral Entry Engineering Test (LEET). You must have a minimum of 60% marks in Diploma to appear for LEET. Incase If you have not given LEET, then try for admission in Private Coll

No, there is no entrance for this course in this college.
Candidate must have passed 10+2 science stream with 60% marks.
Interested candidates can apply online/ offline with the required information. Admission is based on class 12th marks.

Sorry but Mumbai University doesn't offer BMS as a distant learning course. You can do BA/B.Com/B.Sc/MA through distant learning from IDOL.

Total fee for this course is INR 30,000 to 3 Lacs. It all depends on the institute.