Chandigarh University (CU)

You can check the link given below. It has complete information about Chandigarh University admission.
https://www.shiksha.com/university/chandigarh-university-45668/admission.

Yes dear. It is good for your scope. It is the topmost university in India with better faculty and placements. You can get admission there and wish you a good luck.

Chandigarh University offers psychology course for M.A students who have an aggregate of minimum 50 percentage in bachelor's are eligible for the course.
Rest all details of the course are here:
http://www.cuchd.in/basic-sciences/ma-psychology.php

Yes, you can get an admission in Chandigarh University for B.E CSE course through your JEE rank, also you need to have an eligibility percentage of 60 (minimum) to secure an admission. All the course details are:
http://www.cuchd.in/uie/computer-science-and-engg/.

The last date for CUSAT was a week ago so as per CU norms you can't register for it. But, if you want an admission even now then you can apply online through your 12th percentage which should be greater than 60%.

No, Birla Institute doesn't come under Chandigarh University.

Your college fees will be 1.59 lakhs per year and mess fees will be 80K/year.

Definitely yes it is a very good university.

The weightage of the Mains part will be 210 x 0.6=126
Now the board normalization marks depend on the education board under which you took your board exams. If you are from Chandigarh then you have more chances. I think CCA has 40 seats. 35 for Chandigarh students & 5 for other states. Now as you have a good Mains score, that is a plus point and usually most students do not go for Architecture courses. If you are sincerely interested in taking an architectural engineering course, I would say take it. Take those courses only which you truly want to learn.
All the best!