No, DU doesn't offer BSc Psychology, it only offers psychology in BA and Applied Psychology. While these both streams are same but Applied Psychology is mainly associated with more of practical knowledge over theoretical, even I am pursuing Applied Psychology BA from DU and there are more practicals performed over theoretical concepts only. So, choice is yours dear.

Also , being subtle about it ,DU will let you take admission in this stream but since you didn't study it in your high school , there would be a percentage degrade for you to be admitted. For example: You scored 92% in 12th , so whatever college you take ,according to their norms they will consider your percentage with a deduction since its a drawback that you didn't study the subject , it can be 2%, 5% as per the college you choose.

Delhi university does not offer the degree of Bsc in Psychology, but it offers BA degree in psychology at many colleges under delhi university.

the admission to the BA course is based on your class 12th marks, they will in the month of may release the cut off for the courses according to their different colleges.

yes there are colleges, which offer the course even if you dont have psychology in class 12th.