yes, you need a full DGCA class I medical in order for license conversion. You can only fly on the FAA medical if you fly under FATA.
yes, the first and last page of the log book need to be authenticated by the CFI. Obviously each flight also needs to be signed by either the FI or the check instructor or safety pilot etc.
you need to fill something called the CA39 (6 months and 5 yr flying summary) as well as the CA40 if you are giving checks on a multi engine a/c.
the books recommended by the DGCA are vast. but not required. a good starting point would be to look up the "grround studies for pilots" series. there are also dozens of DGCA question banks and notes floating around. get a couple of those and they shall prove to be more than sufficient.
if you are not confident about studying by yourself. there are a whole bunch of people taking classes in delhi and around.