BBA

You can seek admission in BBA even if you are from Biology stream. BBA is a generalist degree open for all stream students. You will not face any challenges except you may be required to put extra effort in Maths & Commerce subjects.

LBS is one of the most reputed school so you need to have a very strong profile. Check here for all the requirements.
https://www.london.edu/education-and-development/masters-courses/mba/apply/admissions-infographic#.V2-s7qBX7qB
The cost for the current session is 70800 british pounds which is equivalent to 70 lakhs approx.

You should opt for BBA through regular college.
BBA distance has no value.

Yes ICFAI is really good option for distance education in BBA. All the best.

You should refer the following link for the same:
https://www.shiksha.com/top-10-bba-colleges-in-india-article-7797-1

It is a good university, but the fee structure is on the high side.

HR comes under hiring employees and developing their skills etc. For doing MBA, you can have any graduation degree.
Now you have a chance to do psychology, better to do it.
As Psychology is useful in all fields.

BBA in Manav Rachna is much better than any other business school as they provide us with different types of BBA courses which suits us the best. After graduation we have a range of job opportunities in various sectors of the industry. Manav Rachna also provides scholarship for students. As students of BBA at Manav Rachna we may also be benefited by its International and National linkages. At Manav Rachna we are also taught subjects apart from the regular management subjects which are an integral part of any business oriented course so as to build a successful professional at the industry.

Chances of getting a good job in Gulf countries after BBA is less in comparison to seeking job after MBA. Also, it will be better if you gain few years experience before seeking a job in Gulf.