Lateral entry

No, you can't get admission in BA because for BA, you need arts in class 12th.

It depends on the college in which you are going to take admission. In private college, the fee is more as compared to a government college. In UPTU, the fess is about INR 96000 for private and 60-80K for the government college. In Punjab, the fee is about 85000 INR in both private and government colleges.

No, NIT does not offer lateral entry. They enrol students for regular MCA programme through NIMCET exam, details are given below:
https://www.shiksha.com/university/national-institute-of-technology-durgapur/exams/nimcet

Hello Prabhat,
Lateral admissions are invariably available in all government engineering colleges in all the states. With thousands of seats not getting filled up in fresh admissions, even majority of private colleges to accept students via lateral admission to shore up strength. So, don't worry.

Of course you are eligible.
Delhi Technological University conducts lateral entry entrance test DTU LEET for direct admissions to the 3rd semester of 4 year B.Tech programmes offered by DTU under the lateral entry scheme. It is a gateway for Diploma holders to get admissions directly in the second year of the undergraduate B.Tech programme.
For more details click on the link:-
https://www.google.co.in/amp/s/www.shiksha.com/university/delhi-technological-university/exams/amp/dtu-leet.

Yes, you can apply for the KDCET for B.Tech lateral entry.

Hello Vishal,
I really do not imagine as to why you desire for admission in Anna University. There are enough well known colleges in Rajasthan itself. However, the situation is very bleak. Almost every state of India has put restrictions on admission. Invariably only domiciles of that state are allowed admission or in some cases NRI & FN's to some extent. So is the case with Anna University. I am afraid you are not eligible unless you can get domicile of Tamil Nadu or your parents are from Tamil Nadu cadre of central services.