Hospitality & Travel

No, currently Delhi University doesn't have this course. But there are many other government colleges and universities that offer this course.

Annamalai University has distant centre there too. Enroll first in university and get in touch to your nearest centre.

You can easily apply for that. You just need to pass 12th for admission.

Please refer to the link provided below for details regarding fee structures of Annamalai University (Distance Education), Tamil Nadu for M.Sc in hotel management. You should be able to download their brochure which should give you access to important details like eligibility, admission process and fee structures of the college.
Link:
https://www.shiksha.com/college/annamalai-university-distance-education-tamil-nadu-tamil-nadu-other-24673

Yes, you can go for it, but you may not get much time to work, as there will be daily classes along with assignments, if you can manage, then you should definitely go for it.

It is already out, and now you can apply for same.

Yes, it is possible to do MBA in hotel management after engineering, if you are much interested.

You should be good in following points
Communication Skills (English)
Food Production (Theory as well as Practical)
House Keeping. Front End Operations. Management Skills. Accounting. Business Law. Nutrition and Food Science.