Upvote:5
You have a number of different questions there, but I'll answer the main one: passports can not be repaired, the only option is to replace it.
Given your timeframe, it will likely be difficult to impossible to get the passport replaced before you leave, so your best choice is probably to try your luck (you'll be OK unless the airline complains) and get it replaced ASAP once you arrive in India.
A discreet piece of transparent tape to hold the cover together would also likely be helpful, just try to position it so that it doesn't cover any details on the bio-data page.