IIT

You should apply for newly opened Iims, nirf ranked management colleges like Lovely, IIHMR, Symbosis, BIMTECH, Jaman Lal Bajaj, KIIT, Dms BHU, DMS Lucknow University, etc. I suggest you to target top 50 B school of India. Also few less known good b schools like IISWBM.

For mathematics, practice is very important. And for that, you need to have a strong base and the formulae on your fingertips. For that, NCERT books would be useful at the early stage. Then, you may prefer the Arihant series for practice which would contain the questions of JEE standard. Tata Mcgraw

Yes, categories ranks are considered while the admission process.

Based on the JEE Advanced marks. Your expected AIR is between 45086-50838. So you can not get any IIT at this rank.

Hello Yash, if I may say so, your statement is a little cryptic. IIT's definitely award degree of B.Tech. But if you can expand and write more in detail about your case and then query, it could be answered in a better way. Best wishes to you.

Hello Nidhi, no one can stop you from learning something. Hence you can definitely learn whatever is being taught to the students of mechanical engineering of any college or university. For this you just need to approach the right teacher who can take you under the wings and teach you the rudiments