1) Try considering IIT Madras as well. Its CS department is one of the most sought after departments. I've got an AIR 60 and I'm applying for it too.
2) Not sure of the finance course(s) at IIT-B. However, I've heard that the undergraduates attend finance and entrepreneurship courses there. You could hopefully audit those courses.
3) Same here, with IIT-M being an add-on.
4) Get yourself registered on COAP first. Then visit,
https://portal.iitb.ac.in