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Yes, it is. You could appear for UCEED exam that is the IIT entrance exam for B. Des courses for all the IITs that offer an undergraduate Design degree. To appear in UCEED, the eligibility is clearing class XII-th in any stream. Why I recommend the UCEED for you is, because as your interest is in Car Design, having a B. Des degree from a technology institute works better, as you get exposure to the engineering side of things as well. After your undergraduate degree you could take up a dedicated Master's degree specialization in Automobile or Transport Design. There are a few private colleges who offer this Master's programme in India besides the Master's programme at NID as well. Internationally too, there are many Institutes and Universities offering dedicated courses in car / automobile design, where the eligibility is not stream specific and anyone clearing Class XII-th can apply.

It is definitely possible, but each fashion institute has its own eligibility criteria. Some may allow admission to a candidate hailing from any educational background while others would prefer a candidate who has done graduation in design field itself. You will have to check the criteria of all the colleges that you are interested in. For example: Pearl Academy will not consider your application for the Masters programme, but NID and NIFT would accept the application as the admission criteria is solely dependent on how you perform in the entrance exam. You can find a rank wise list of top design colleges in India by clicking on this linkhttps://www.shiksha.com/design/fashion-design/ranking/top-fashion-design-colleges-in-india/94-2-0-0-0
Good luck.

ISD students register online, generally two to three months prior to the start of each semester.Fall: registration opens in June, classes begin in September.
Winter: registration opens in October; classes begin in January.
Spring/summer: registration opens in February; classes begin in May.
Requirements for online registration:
Students must be admitted into ISD either as degree-seeking students or non-candidate-for-degree (NCFD) students. Information on the admissions process is available here. NCFD students may take up to two courses without being formally admitted into a programme. Students must have a UMID. Contact us if you do not have this number. A high-speed internet connection outside of a corporate firewall is required to access online course lectures.
Steps for online registration:
Read over the online graduate courses for the upcoming semester. Be sure to take note of any prerequisites and system requirements. Using your pre-approved plan of study, identify the classes for which you would like to register. Select the "Register Now" button, which then prompts you to enter your uniqname and UMID. Complete and submit the registration form. If you have any questions or need assistance, please contact your graduate coordinator or ISD.

Hi Ashish,
According to my personal view, every branch has its cons and pros. If you have a thought of Computer Science & Engineering/IT as best branch then, my friend you are wrong :)
CSE/IT provides extremely competitive environment. Nowadays, everyone is interested in coding whether it is basic C or Java. So if you have faith in yourself and love to code because of your passion (not because of the fact that CSE/IT gets the highest package offer to this branch in almost all engineering colleges/ universities), you should go for it. Mechanical Engineering is one is the toughest of all branches. If you are interested in Robotics or how machines works, then this is your way to rule the world :)
Next is Electrical and Electronics engineering, and it is a branch of Science which deals with the applications of electricity, electronics, and electromagnetism. Civil Engineering is all about conceive, design, build, supervise, operate, construct and maintain infrastructure projects and systems in the public and private sector, including roads, buildings, airports, tunnels, dams, bridges, and systems for water supply and sewage treatment. Many civil engineers work in planning, design, construction, research, and education. If you want to go for public service related jobs or passionate about serving to nation then this is best for you. Starting Salary is low as compared to other branches when we take private job into consideration. If you are good at chemistry in class 12th, it is doesn't mean that Chemical Engineering is for you. It uses principle of applied physics, life sciences (microbiology and biochemistry), applied mathematics and economics to efficiently use, produce, transform, and transport chemicals, materials and energy and also little bit of chemistry.
At last, I would suggest you to see your interest level in different field by small research over NET about what really you are passionate about. First introspect yourself and then take any decision. All the best.