CBSE 12th

Yes, if you wish to do your class 12 along with your graduation, then you can definitely complete your 10 + 2 from NIOS board. NIOS is one of trusted open schooling platform and if you pass the 10 + 2 from this is schooling system, then you will be eligible to use your marksheet for any examination as the institute is Government of India recognised.

If you fail in more than one subject, you can't give compartment exam, so you have to reappear for the 12th board exam. You cannot reappear for only three subjects. You have to appear for all the subjects.

You might get little problem if you have not studied Physics and chemistry in class 11th and 12th as the first year of BCA would have Engineering Physics and Chemistry, however if you are so keen into doing it then I can suggest GL Bajaj for it. You can pursue your BCA then MCA for masters from here as MCA placements are better than B.Tech students and yes there are options for specialization so you can opt for AI and ML too.

Yes, you can apply for B.Sc. Computer Science (Hons.) at DU with your commerce background, only if you studied Mathematics in Class 12. 

Subjects you can take in CUET for DU:

1 Language (from CUET List A)

Mathematics or Applied Mathematics

Any two domain subjects from Section II (Computer Science, Physics, Chemistry, or Economics)

If you didn't have Maths or Applied Maths in your 12th class, then you can't choose it in CUET or apply for BMS or BBA at Delhi University. They clearly ask for it as a compulsory subject. Even if you try selecting it in the entrance, it won't count. One option is to take Maths later through something like NIOS and then try next year. But without having it in 12th, you won't be eligible for those courses at DU.

To get admission in Bhagalpur College of Engineering, you must have passed your 12th with 45% marks.

But not only 12th, you have to give JEE Mains or BCECE. On the basis of the results of these exams, it may be decided that you can get admission in this college or not.