IIIT

Kindly visit this link for the detailed information:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

You can try for GFTI colleges, new IIITs and eastern NITs or NIT Srinagar.

Yes, you can get NITs / IIITs if you have scored above 75% in class 12 or are in top 20 percentile of your board and scored good rank in JEE Mains.

Frankly speaking, IIITs are well known for information technology courses. It's quite tough to differentiate between these institutions. However, to help you I am attaching a link that will give you all the required information about not only these institutions but all the colleges in India which are offering this branch for studies.
https://www.shiksha.com/b-tech/ranking/top-computer-science-engineering-colleges-in-india/46-2-0-0-0
Best wishes to you.

You have no chance in JoSAA, but you may get through CSAB rounds for new IIITs.

You have no chance in JoSAA rounds. Better to try for new IIITs, GFTI, lower NITs in CSAB / special round.

You can check the previous year's cut-offs here:
https://josaa.nic.in/Result/Result/OpeningClosingRankArchieve.aspx
There you may get an idea of your selection. Also, keep track of the important dates of JoSAA counselling.

If you have scored 67% in HSC board then you are not eligible for JoSAA counselling.

Please use the college predictor tool by Shiksha. It will help you to predict the college according to your rank/percentile:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

No, you are not eligible.