IIT

Yes, they do offer this course.

Yes, it offers genetic engineering.

Apart from IITs there are NITs, BITS Pilani, AMRITA, VIT, SRM, etc. are some of the top colleges.

You can study Bachelors in engineering at IIT. But if you have completed your degree in any other colleges, you can study Masters in IIT's like you can pursue MCA from IITs after completing BCA.

Some of the best options comprise:
BITS Pilani
PSG College of Technology
Sardar Vallabh Bhai Patel Institute of Technology
VIT
SRM University
LPU

You need to get 90% in general category.

BITS - Pilani is the best non IIT/NIT University for B.Tech in mechanical engineering with placement records equivalent to legacy IITs.

1. BITS – Pilani.
2. VIT.
3. SRM University.

IIT and NIT do not offer undergraduate engineering degree course on embedded systems. All electronics and CSE graduates have embedded systems paper/subject in their syllabus. IITs are the best institutes for pursuing M.Tech embedded systems. IIT - Delhi is the best option for the subject.