Mechanical Engineering

To secure a seat you need to have a rank below 5000. Further, the ranks are admitted according to the financial attraction of the course and the time of admission.

Andhra University
GMRIT
JNTUK
KITS
GPREC.

The definition of best varies from person to person and is influenced by personal inclinations and one's strength. For example, some people might go with a regular stream like CS mechanical, Electronics and some might go with not so common branches like robotics, petroleum, and mining. Opt for the engineering branch that you love the most and are most passionate about and analyze the future work conditions, scope and growth prospects before taking admission
You can also opt for psychometric test comprising of Aptitude, personality and career interest to clear confusion.

Top Engineering Colleges in India Offering B.Tech lateral entry are mentioned below:
1) Delhi Technological University
2) Manipal institute of Technology, Karnataka
3) Netaji Subhash Engineering College, Kolkata
4) Jamia Millia Islamia
Regards.

There are over 5 Distance/Correspondence B.Tech colleges in Kolkata. Please refer to the link provided below for a detailed list of these colleges. You should be able to navigate to a college of your choice using the filters on the web page. Regards, https://www.shiksha.com/b-tech/colleges/distance-correspondence-b-tech-colleges-kolkata.

Hi Vicky,
It depends on your priorities. You have the option of getting a job and pursuing your higher studies with some experience. Or you can choose to pursue your higher studies right away. With higher studies, your job prospects increases and I would recommend that you do pursue the same though.

Your rank is very good. You will easily get this college with your desired branch. Try JEE college predictor to get more information and insight about college & branch which you can get on the basis of your rank:-
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=4067795.

No, it totally depends on your interest as there are many career opportunities depending on your knowledge in every course. And doing BA after your diploma is good . You have many opportunities after doing BA so there is lot of things to do in your field do you can go for this.
Good luck.