Please find the answers to your queries:
1) Almost all IITs offer some Integrated M.Sc course or the other these days. However, full fledged programmes have been fairly regular for a long time now in IIT Kanpur and IIT Kharapur.
2) There are no other exams to enter the integrated M.Sc courses except the IIT JEE.
3) There are no courses in the IITs in astrphysics as yet. However there are certain centers from which you could opt a few courses in your curriculum which would enable you to obtain opportunities for further studies in astrophysics.
Kishan S Srinivas