National Institute of Design, Ahmedabad

Yes, you can apply.
NID Entrance Exam 2017 Eligibility Criteria for BDes Admissions:
The eligibility criteria for NID entrance exam states that candidates who have cleared or who will be appearing for their qualifying exams under higher secondary (10+2) from any recognised board such as CBSE, IB, ICSE or their equivalent in the year 2016-2017 can apply for the exam.
Age Limit: The upper age limit to meet NIDs eligibility criteria is 20 years (on or before June 30, 2017). This criterion is relaxable by 3 years for reserved category students.
General Category/Overseas Category: Candidates born on or before June 30, 1997 are not eligible to apply.
Reserved Categories: Candidates born before June 30, 1994 are not eligible to apply.

Yes, National Institute of Design offers B.Des.and M.Des. in product design.
Eligibility criteria for the same:
B.Des - A General category candidate should not be more than 20 years old and this age limit can be extended to another 3 years for Reserved categories. The candidate must have completed or must be completing his/her Higher Secondary education or equivalents like AISSCE/IB/ISC.
M.Des - A General category candidate should not be more than 30 years old and this age limit can be extended to another 3 years for Reserved categories. The candidate must have completed or must be completing his/her Bachelor's degree or an equivalent Diploma. Degree in design, engineering, technology or architecture, or a recognized Diploma in Design are the most preferred academic qualifications.