Karnataka State Open University

Unfortunately, due to the time gap, you cannot pursue 2nd PUC.

No, you have to complete your degree in one go only.
Good luck!

Yes, you can pursue degree from some private institute as government may not allow admission. But, you can try to take some private institute.
Good luck.

You have not mentioned that what kind of science course you want to pursue. However, let us assume that you want to do the 12th in science stream. You can do so, but you cannot avail the benefits of both the courses at same time. Means the benefits of Diploma and 12th qualifications cannot be claimed at the same time for employment and higher education purpose. Therefore, it is better to complete your Diploma first, and other qualifications can be obtained later.

Yes, you are eligible to appear for the exams.