JEE Advanced

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.

You can try for IIT. You will be benefited by the reservation quota and you might get a lower branch in one of the newer IITs. So, try for these but don't be dependent only on IITs. Go for counselling for other colleges too.
Good luck.

Your rank will be around 8000 .I think at this rank you will get a seat in IIT in spot round counselling. My brother got 146 and he is currently in IIT, Dhanbad (mining machinery branch).

In which state are you looking for?
See for the branch, try to choose the field in which you love to work and you have very much interest in it with curiosity, it will be very much beneficial to you than any other fields.
Best of Luck and Work.

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.shiksha.com/b-tech/articles/jee-advanced-2016-expected-cut-off-released-blogId-12688
and
https://www.shiksha.com/b-tech/articles/revised-cut-offs-for-jee-advanced-2015-blogId-11007.
Hope this helps! All the best!

Yes, JoSAA conducts counselling for IIT/NIT and GFTIs. Candidates are admitted through JoSAA in IITs and NITs.

JEE Mains counseling will start after declaration of JEE Advanced result. So you should wait for the announcement for advanced result. Admission in IIT/NIT/GFTI will be done through JOSAA. You have to fill choices of your desire colleges and branches. There will be 5-7 rounds of counseling. For more details go to www.josaa.nic.in.

If you are looking for a B.Tech course, you need an AIR under 100. If you are looking for a dual degree course in computer science, an AIR under 200 should suffice.

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.