Punjab

Yes, Guru Nanak Dev University does offer distance learning programme for a variety of degrees and branches starting from M.Sc, B.Com and going on till LLB too. You can find the relevant details for the same on their website.

Admission is based on the marks obtained in class 12th.

No such course is available. https://www.shiksha.com/college/goswami-ganesh-dutta-s-d-college-chandigarh-22606/courses/bsc-bc.

Yes, you can get B.Sc(Hons.) maths if you have filled PCM.

For this, you can contact to the college authority directly as in my knowledge there is no such provision of getting directly in to the 2nd year of BCA in this university.

Yes, as per the description on shiksha.com and their website, the BCA course is available there. Please, visit the links for more information:
https://www.shiksha.com/university/Punjab-university-Patiala-24117/courses/bca-bc
http://www.punjabiuniversity.ac.in/index.aspx.

Thapar Institute of Engineering & Technology offers undergraduate programs leading to a Bachelors degree in Engineering and Technology, Post graduate programme in Engineering Science and Arts, and Doctoral programme.

Can you please specify the name of the college where you wish to apply?