Electrical Engineering

No, you don't need to write an entrance exam to join S. Veerasamy Chettiar College of Engineering & Technology for BE in electrical & electronics engg.
Admission is based on TNEA counselling on the marks obtained in 10+2/Diploma
For more details check the link below :
https://www.shiksha.com/b-tech/course/b-e-in-electrical-and-electronics-engineering-s-veerasamy-chettiar-college-of-engineering-and-technology-tirunelveli-237715

No, IIT does not take admission after 3 years since IIT does not provide lateral entry and hence you can't get admission in IIT IIIT, etc. You can get admission in lateral entry in private colleges.

Dear Harikrishnan,
Thank you for showing interest in Ganga Technical Campus. Admission in B.Tech Lateral Entry is possible if the student has completed his Diploma in Engg/Technology, duration of 3 years or more from Haryana State Board of Technical Education or its equivalent/B.Sc degree from a recognized university as defined by UGC and pass in class 12th with mathematics a a subject. For more details, visit us at:
www.gangagroupofinstitutions.com

B.Tech/BE through lateral entry (direct second year admission) will be best for you, you have to appear for LEET examination to get a good college.

So, you can get a lateral entry admission for engineering.