Colleges in Kochi

It is 2 years in general but depends particularly on the institute/university. I would suggest IGNOU.

You can apply to XIME and Xavier University using your CMAT score and the cutoff is around 80 percentile. On the other hand XLRI accepts only XAT and GMAT scores.

Jaipuria Institute of management would be far better option. They are among the top b-schools of India. Check NIRF rankings by Ministry of Human Resource Development.
2020 rankings: https://www.nirfindia.org/2020/ManagementRanking.html
2019 rankings: https://www.nirfindia.org/2019/ManagementRanking.htmlThe standard of education is high because the faculty members are highly qualified and extremely helpful. Good companies like Amazon, HUL, etc. Come for placements. So, Overall, Jaipuria is far better.

It is possible. There are several universities that offer B.Tech through correspondence. Go ahead. It is possible. Wish you luck in your future endeavors.

Once I have been there in XIME, Bangalore and that was quite impressive and had good placement records. I am not aware about its Chennai campus. So please check the details from the campus or their website. In case you are looking BITM Pune then I would suggest there are many other colleges which are better than this in Pune area.