Engineer

Tata, Mahindra, Reliance, Lohia, Torrent, Thermal, Godrej, Larsen Turbo, Kirlosker, General Motar, Siemens, etc.

Yes, He/She is eligible for Bachelor of education.
https://www.shiksha.com/teaching-education/colleges/b-ed-colleges-india
Here is the list of the B.Ed college in India

Metallurgical Engineering - Metallurgical Engineering is an important branch of engineering that deals with the study of the physical and chemical behavior of metallic elements, their inter-metallic compounds, and their alloys. Physical metallurgy needs problem solving skills.

Yes, you can be eligible for GME course at GEIMS, if meeting the below eligibility criteria.
1. Educational qualification : Graduation in BE (Mechanical) Engineering / Naval Architecture
2. Percentage required : Minimum marks of 50% in final year and with Minimum 50% marks in English language at 10th OR 12th OR in Degree Exam.
3. Age : Not more than 28 years (as on course commencement date).

Yes, of course if you have completed a Diploma, then you are eligible for lateral entry in B.Tech. Yes, you can complete your B.Tech in distance mode or by part-time B.Tech course.

Yes, through lateral entry. But not every college allows lateral entry.

I must confess that you have raised a very interesting query. Unfortunately asked for information is not available in their brochure though the university claims that they have different avenues and facilities for different sports. I am attaching a link that will give you required information about Amity Kolkata. Else you can ring them up for further details.
https://www.amity.edu/kolkata/
Best wishes to you .