Engineering Entrance Exam

No, it is not necessary to take coaching. Refer study material and use online sources like YouTube. For Physics, you can use YouTube channel Physics with Pradeep at free of cost. Use google for free online mock tests. If you work hard you do not need such coaching. Even I cracked JEE Mains without coaching.

There is one government college, Chandigarh College of Architecture mainly known as CCA. You can check the cutoff list of all categories on the official website of Chandigarh College of Architecture.

For the SNU SAT, the syllabus is the same logical reasoning that you have learnt for the other exams such as BIT SAT.
Coming to Apt,
For B.Tech, it is physics & mathematics that you have learnt in class 11th and 12th.
Trust me, there won't be any negative marking, so as an experienced one, I suggest you to attempt everything.

Chandigarh University
Chitkara University
Amity University
You can try for government colleges too. If you get them, it will be better.

Chances are low. You won't get admission in any good college.
Try next year or try for private colleges.

As per the score that you have got in JEE Mains exam, chances for getting any IIT are nil but you can get some other good engineering. The maximum cut-off score is 81 for general students and you have 83. Participate in the counselling process of private as well as government colleges. You will easily get admission in a good college.

Your chances are fine.
Cutoff of Punjab Agriculture University, Ludhiana exam will be evaluated on many factors some of factors are mentioned below:
Number of Candidate appeared for the Punjab Agriculture University, Ludhiana exam
No. of candidates passed the exam
Difficulty level of question paper
No. Of vacancy for this exam
Cutoff may also be sectional wise

You need to get 75% marks from Physics, Chemistry and Maths for general category for getting into IIT, Gandhinagar.

Hello Himanshu,
You can only get admission in PSG through state board counselling, they don't give admission based on JEE Mains score.