Vellore

Well, it is a tough question to answer. But for CSE I will suggest Vellore over DTU.
DTU admits over 300 students in CSE branch every year. Competition for the job is quite tough.

Yes, there is no doubt that VIT is more preferred over Vellore, as it is their main campus. They actually have an advantage that Bengaluru is the largest Technological hub of India, so they have a great location advantage, and they have a very good faculty base, as I was told by a relative of mine who studied there. So, I would recommend to go for VIT-Vellore.

You can appear VITMEE (Vellore Institute of Technology Masters Entrance Examination).
To know more details about this exam please visit the link below: https://www.shiksha.com/engineering/exams/vitmee.

Sorry, you need to qualify your CAT exams to get admission in VIT Vellore because without examinations or entrance examination you cannot go for further studies in VIT Vellore. So please give exam, qualify that and then take admission.

Yes they have M.Tech in design engineering course.

Previous year question papers will help you best to understand the pattern of the paper.

There is no entrance examination.
Candidates who have studied in Regular, Full time and Formal Education are alone eligible to apply.
Candidates whose date of birth falls on or after 1st July 1998 are eligible to apply.
Application form will be available from the first week of April 2018.
The Admission will be on the basis of the performance in the qualifying examination (10+2).
The candidates are short-listed based on their merit will be called for the counselling.