IIT

Core courses are often much helpful for future education. You can learn in new IIT. After all it is an IIT though it is new. Other course often leads to problem in our further education.

Yes, According to Josaa Counselling Process, It is NE to score 75% in HSC in order to get admission in IITs NITs and IIITS for B.Tech.
Best of Luck and Work.

AAT exam is for you to get into IITs to pursue a course in Architecture. There is, of course, not any compulsion if you are not interested in B.Arch, whereas, JEE Advanced is for students who are interested in pursuing B.Tech from IITs.
Hope this was helpful.

Kindly use the college predictor app. Hope this will resolve your query. https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
All the best.

To qualify for IITs, you have to give JEE Advance and secure a good rank. 87 marks will not give you good rank. So, you cannot qualify for any of IITs.

Find out which rank will you get by entering your JEE Advanced marks in the link given below-
https://www.shiksha.com/b-tech/resources/jee-advanced-rank-predictor
Once you know the rank, check out the colleges cut-off ranks for mechanical branch and compare with the following scores:
https://www.shiksh

If you are general or OBC then chances are very less but if you are from SC or ST category then you may get a seat in IIT. Although we can only make predictions so wait till result.

Sorry but your marks are not enough that can help you to get a college because above 300 marks are considered for admissions in B.Arch. So, you should try for any private college.

It would depend on what you further aspire. You can take higher studies like M.Tech in various fields or branches in petroleum, or take up management courses like full-time MBA or executive MBA or so.

I think you will get admission in an IIT. I am not sure about girl’s quota implementation from this year. But if they have implemented girl’s quota then you will get a good branch as well.