Exam

Yes, you are eligible to give the entrance test. You must get passing grade to sit for the entrance exam.

Hello,
I hope you are doing well. You can check the list of colleges including NITs and IIITs that you can get according to your marks and rank. Use Shiksha JEE College Predictor:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=

Yes, they do offer hostel facilities.
https://www.nift.ac.in/mumbai/Hostels

Online registration starts in the last week of October and ends in the last week of December. GAT is held in the last week of January.
You can refer to this calendar:
https://nift.ac.in/admissions

Better than just selecting only one college, I would like to suggest you appear for CAT, MAT and CMAT. This will open the doors of many other good institutes for you.

Hello Priyansh,
Even if this query is pretty old, it still has relevance today. I would advise students not to get frustrated by low scores. Keep on trying till all doors are not closed. Last door is perhaps called spot admission. Wait till this too gets exhausted.

Go for the JEE Mains counselling and check whether the name of the college is present in the Mains list and then if it matches your rank, go for it if your rank is nearby. Wait for the last round of counselling. Do not leave a seat.