Design

First, you must clear out what you want to opt. You want to go for BF.Tech which is Bachelor of fashion technology or do you want to go for engineering in design?
If it is BF.Tech, then you can apply for NID or NIFT where most courses are offered.
If it is the latter one, then other than NID, you can find courses like automobile designing, animation/graphic designing etc in IITs all over India.
P.S. both streams require class 12th maths.

Just follow your dream and interest. Because if you have no interest in engineering then you won't be able to study properly and might regret later for wrong choice. If fashion is what you are interested in then you should definitely opt the same as this will be the deciding point of not only your career but your whole life.

NIFT entrance exam is valid only for the NIFT colleges and it's branches in overall India.

Yes, you can still fill the form for NIFT until 29th of December 2017, which is the last date and the exam CAT and GAT both commences on 21st of January 2018.