Pune

Yes, they take CMAT scores accepted through DTE rounds. 96 percentile is the general cutoff.
You can check the below mentioned link:
https://www.shiksha.com/getListingDetail/30895/institute/college-Department-Of-Management-Sciences-Savitribai-Phule-Pune-University-Pumba-Pune-India

For mechanical engineering :
https://www.shiksha.com/top-mechanical-engineering-colleges-in-pune-rankingpage-48-0-0-174-0
For automation engineering :
Planet Automation Systems
Chintamani Nagar, Kondhwa Khurd, Pune
09850396034
Prolific System ans Technologies pvt ltd
www.prolifictraining.com
Office No. 3

Padmashree D Y Patil College of Architecture, Vidya Pratishthan's College of Engineering (VPCOE) and Bharati Vidyapeeth College of Architecture are the top colleges for B.Arch in Pune.

There is a Common Entrance Test (CET), only through CET the admission to such colleges shall will given, which is conducted by the Maharashtra Government. It's more like an entrance exam.
For more proper information, follow this link: https://ilslaw.edu/admissions-by-central-entrance-test/

ISB&M Nande is a much more better option than BIMM not only on the academic basis but for the further individual enhancement which leads to over all development of a candidate. Regarding the infrastructure of the campus, it is situated in one of the best location of the Pune with good accommodation

Among these two, MITSOM is better option. You could also try Balaji group.

Following are the best colleges as per the brand value:
1) IIEBM
2) St.Francis
3) Indira

No , Mathematics is an important subject in BCA, so you are not eligible for it