Diploma

Yes, you most definitely can pursue a course in B.Sc. (Interior designing) after completing a diploma in interior designing.

Yes you can get admission by giving NATA entrance exam.

Yes, you can pursue BBA provided you have done your DBM from a recognised institute. Usually to get admitted in BBA courses the requirement is 12th pass or an equivalent course from a recognised institute. So a lot depends upon where you have done your DBM from.

Hello Adarsh,
The cutoff varies every year according to the number of students applying for examination and number of seats available. It would be advisable that you get in touch with college authorities/official page directly for accurate cutoff which can help you in figuring out the appropriate col

Many colleges are quite strict about the age criteria, you need to check the eligibility with the desired colleges for admission in diploma.