Diploma

Yes, you can apply for the course.
Diploma in Culinary Arts:
Eligibility criteria: Students who have passed 10+2 board examination (any stream- Science/Commerce/Arts) are eligible to pursue this course. Minimum aggregate marks criteria may apply in case of some institutes (generally around 50% marks).

It is the colleges which require compulsory AICTE approval, so if you are directly studying at the university you do not need to worry. On the other hand, if you are studying in a college affiliated to some university and your course is not approved by AICTE, then it can create lot of problems.

Diploma is not considered as equivalent to graduation and minimum eligibility for any MSc course would be graduation.
http://online.ideunom.ac.in/newone/index.html

Management quota is an option for you. The admission under management quota is processed in the office of the BMS Educational Trust. The Council of Trustees decide the tuition fee for various courses under management quota and seats are allotted by the Chief of the Admission Committee. The BMS Educational Trust advertises regarding the admission and the application forms are issued to the aspirants in the month of June / July of every year.

Hi
Yes, you can take admission after dropping. You will have to go through the prospectus of the course provided by the college to know more about the admission procedure, fees, eligibility etc.

Visit - http://dypatildistancemba.com/OnlineMBA/

Follow this link to get a comprehensive list of all the colleges:
https://www.shiksha.com/medicine-health-sciences/ranking/top-medical-colleges-in-india/100-2-0-0-0

Hi
You will have to go through the prospectus of the course provided by the college to know more about the admission procedure, fees, eligibility etc. https://www.shiksha.com/medicine-health-sciences/neet-exam-dates