I would say both the courses are good in their own way. If we think from job point of view I would say bcom will be more better as it will open your doors for so many reputed courses. Afterwards your salary packages will boost up. BBA is more inclined

Here are some of the colleges for BBA you can prefer.

Alliance School of Business, Alliance University

Institute of Business Management and Research- International Business School

KLE Society’s Degree College

St. George College of Management, Science and Nursing

St Pauls Institute of Communication Education

Good Luck!

Opportunities for integrated bba mba are :

Business analyst or strategist

Business development analyst, associate, or manager

Director (of a department)

Entrepreneur/founder

Financial analyst

Management consultant

Marketing associate, analyst, or manager

Portfolio manager

Project, product, or program manager

Opera

Top colleges

Ahmedabad University, Ahmedabad

Jindal Global Business School

Shaheed Sukhdev

No, you are not eligible for GATE 2018 as it requires 4 years graduation course like B.E/B.Tech/ B.Sc etc. Also, you can check the complete Gate eligibility below -

Gate Eligibility 2018

Or you can also try for below provided courses once you complete B.Sc Electronics -

MCA

M.Sc in

There is no restriction. In some colleges Maths and English are must subjects for BBA studies while in others there is no as such requirement.
So you can easily take up BBA. There are entrances for this course in various institutes like
DU JAT
SET BBA
IPU BBA
Christ BBA

Yes. The Distance Education Council (DEC) has accorded recognition to the ICFAI University, Tripura for its distance education programs. The relevance of DEC recognition is limited only for seeking jobs in central government. So yes it  valid for government jobs.

All the best!