If we want to pursue B.Ed, do we have to choose the subject on our own or is it given on the basis of our graduation degree/subject?

At the time of training, you will be teaching till class 10th student if you do graduation only, and if you have done post-graduation, then you can teach class 11th and class 12th as well (of your master degree from Science, then you can teach to Science students of class 11th and class 12th.)