Diploma

Hi Jafar,
If you're choosing to do B.Tech through lateral entry, then it's imperative that you should do a Diploma in that respective branch. Shiksha has created a list for the Diploma courses offered by various colleges, that you can avail:
https://www.shiksha.com/engineering/colleges/diploma-mechanical-engineering-colleges-india.

For admission in NIT, candidate must appear for JEE Mains. Unless you appear for the entrance exam you won't get admission in National Institute of Technology. The eligibility criteria for entrance exam is 12th with PCM as the subjects.
Apart from IIT and NIT, most colleges in India offer lateral entry for diploma students. These colleges do reserve some seats for lateral entry students.Diploma holders are not allowed for JEE Main admissions but they are allowed for JEE Advanced. So they can write the JEE Main exam just to qualify for JEE Advanced.

According to the the ratings and reviews, it is a good institute for PG courses.

Hi, Please find the link below:
https://www.shiksha.com/mba/xat-exam.

The syllabus would include engineering mathematics, electrical technology, computer application, engineering mechanics, strength of materials. The syllabus remains the same irrespective of the branch.

New Zealand PR is based on the shortage skill list listed in NZ immigration web site. Supply chain management seems to be not there. However that does not mean you should not go for this course. It all depends upon what UG programme you did and if you could get a job related to that studies you may be able to get PR. Again PR depends upon you age, qualification, experience and IELTS marks. If you have good (8 band IELTS then it is possible.

NCHMCT is for only UG course in hospitality.