Commerce

You can give various exams after 12th class commerce such as SSC, banking, railway, state level examinations, postal department and defence sector.

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)

Yes, you can get admission in government colleges for sure. I would suggest you to crack CUET as through CUET you can get admitted to some of the best government universities.  

Of course, Psychology is a subject for which courses are offered in various types. If you have passed Class 12 with the Arts or Commerce stream, you can opt for a BA, BA (Hons.), BVoc, or even a BBA in Psychology. If you really want to study Psychology, it is not mandatory to be a Science student. Psychology is a vast domain, and it does not apply only to Clinical Psychology.