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TNEA (Tamil Nadu Engineering Admissions) is conducted by Anna University, Chennai for admission to various engineering colleges in Tamil Nadu. To be considered as a "local candidate" for TNEA you must satisfy certain eligibility criteria set by Anna University. As per the TNEA eligibility criteria, a candidate can be considered as a local candidate if he/she meets any of the following criteria:
Candidates who are Native of Tamil Nadu (i.e. who have studied in Tamil Nadu) or who have completed their qualifying examination from an institution located in Tamil Nadu. Children of Central Government Employees, who have been serving in Tamil Nadu continuously for the past five years at the time of submission of application, and whose parents' native is not Tamil Nadu. Children of employees in Public Sector or Government recognised institutions who at the time of submission of application for admission employed in Tamil Nadu continuously for the past five years. Based on the above eligibility criteria, if you have completed your 10th, 11th, and 12th standards in Tamil Nadu, then you can be considered as a local candidate for TNEA. However, if you do not meet any of the eligibility criteria mentioned above, you may not be considered as a local candidate for TNEA.

The admission process for Amrita Vishwa Vidyapeetham is based on the rank obtained in the Amrita Entrance Examination Engineering AEEE as well as the availability of seats in the preferred branch and campus. The cutoff ranks for admission to different branches and campuses may vary from year to year based on the number of applicants and their performance in the AEEE. As per the cutoff ranks of AEEE 2021, the closing ranks for admission to CSE-AI and CSE at Amrita Amritapuri campus were 1015 and 1385, respectively. Since your rank is 1586, it may be difficult to get admission to CSE-AI or CSE at Amrita Amritapuri campus based on AEEE rank alone. However, please note that the cutoff ranks are subject to change every year, and other factors such as category, availability of seats, etc. May also come into play during the admission process. Therefore, I would suggest you to contact the admission office of Amrita Vishwa Vidyapeetham or visit their website for more details on the admission process and the cutoff ranks for the current year.

The ranking required to secure admission in mechanical engineering through KCET (Karnataka Common Entrance Test) varies depending on the college and the category of the candidate. However, generally speaking, a rank below 10,000 can be considered a good rank for admission into mechanical engineering in a reputed college in Karnataka.
As for the fees structure, it also varies depending on the college. However, for the academic year 2021-22, the fees for mechanical engineering in government colleges in Karnataka was around Rs. 25,000 to Rs. 35,000 per year, while for private colleges, it can range from Rs. 50,000 to Rs. 2,50,000 per year. It is important to note that these are approximate figures and can vary from college to college.