Gujarat

To pursue BBA in Ahmedabad, GLS is the best option. To pursue BBA in Baroda, MS University is a good option.

You should appear for JEE advance, then you will be able to select NIT - Surat. But, your marks are good in JEE Mains exam, so you get selected for NIT - Surat.

Check these links-
https://www.shiksha.com/bba/colleges/bba-bms-bbm-bbs-colleges-in-ahmedabad
http://indiacollegeinfo.com/top/BBA_colleges_Ahmedabad

In terms of placements, Shanti B-school is better than St. Kabir. The placement depends on your own career eligibility, academic performance and communication ability. You can improve that as well, then placement is not going to be a problem for you.
All the best.

You may refer to the following link for a list of the best colleges to pursue chemical engineering in Gujarat: https://www.shiksha.com/top-chemical-engineering-colleges-in-gujarat-rankingpage-52-0-109-0-0

CMAT score is accepted at many management colleges including Adani Institute of Infrastructure Management (AIIM) - Ahmedabad, Amrut Mody School of Management-Ahmedabad University, BK School of Business Management, Gujarat University, Maharaja Sayajirao University of Baroda, Vadodara etc. Depending upon your choice of stream in MBA and city, you can find out which colleges take CMAT score and its cut-off as well from their respective college websites.

Many private universities in Gujarat and other states offer integrated engineering and MBA.

It depends on the college you are going to choose. All the colleges have different criteria and some private colleges give direct admission, but CMAT is compulsory for the same.