GATE

Since Chemistry is a division of Science, hence, you can definitely apply for the examination. For more details regarding GATE examination, please go through the following website:
http://www.iitk.ac.in/gate/gate2011/eligibility.php.

Hello Kishan,
Yes, of course. You can definitely and surely seek admission in Aeronautical engineering through GATE. Just select proper subject. That's all!

Yes, you can appear for GATE. To be eligible for GATE exam, you need to have Bachelors degree in engineering/technology. Masters degree holders in any branch of science/ mathematics/ statistics/ computer applications or equivalent and those who are in the final year of such programs.

Instead of country, look for the institution. Generally, US have better institutions for this branch.

You can go for any good coaching institute or if you are confident enough to clear it by own then prefer self-study with a proper time table & as you know there are many books, websites, etc., available for mock tests, so you must add it to your practices daily.
Good luck.

IIT-Kharagpur and Delhi are the best IITs and National Institute of Technology-Tiruchirappalli, and National Institute of Technology-Karnataka, Surathkal are the best NITs M.Tech in mechanical engineering.

Check out the article below for all the details : https://www.shiksha.com/engineering/articles/how-to-prepare-for-gate-in-4-months-blogId-15035