I don’t know if there are any married (and thus hair-covering) Orthodox women who post here, but I’ll take a crack at it. (FWIW, I’m a single Orthodox woman. Are there any married frum female posters out there? Other than Zahava424, who’s also single, the other Orthodox posters I’ve met are all guys.)
Orthodox Jews don’t want to use anything that would have been used in a non-Jewish religious ceremony, as using it might be construed as participating in/condoning such a ceremony. I don’t know exactly under what situation Indian people would be cutting off their hair in a religious ceremony (the article in the New York Times doesn’t go into it), but if they’re cutting it off as part of a religious service, that hair wouldn’t be usable for Orthodox Jews. Intention is key - if (let’s say) they’re cutting it off because the ceremony requires them to have very short hair, but the cutting itself isn’t intended as part of the worship, the hair is probably kosher, although I’m definitely not an expert in these laws. If the part of the actual ceremony involves hair-cutting, then it’s probably not kosher.
Cranky, that sounds like a snood. Different women feel comfortable exposing more or less hair, and some certainly leave several inches worth of hair uncovered in front of the snood (ie they have the hair down their back inside the bag, with the elastic rim sitting several inches back from the top of their forheads.) It’s a personal call, and there’s no one standardized answer as to exactly how much hair one must cover (within a certain range.) Most women I know don’t wear them in public, exactly, they’re more for casual wear among other Orthodox Jews. (Coworkers have asked me why a fellow Orthodox female grad student is wearing a giant sock on her head, which I think explains pretty well why I wouldn’t want to wear one to work when I get married, no matter how comfortable they’re supposed to be.)
In general, modesty is defined in different ways by different cultures, and I think it’s hard for people of one group to understand the rules of another; think of African tribal groups in which the women used to walk around bare-breasted, but would always wear long wrap skirts - showing any leg beyond the ankle was considered scandalous. From what I’ve seen, Indian women have very strong standards of modesty but see nothing racy about baring their midriffs. Women wore incredibly low-cut gowns to Victorian balls, but one couldn’t acknowledge that they even had legs, let alone see them. Among Orthodox Jews, married women walking around with their heads uncovered is considered immodest; covering is correspondingly modest behavior, irrespective of what you’re covering it with. Having grown up watching women-with-wigs all the time, trust me, you learn to tell the difference, even with the really, really expensive ones - people within the Orthodox community can tell without thinking about it. Conversely, most (non-Jewish) people seem not to notice pretty bad wigs, probably because they’re not looking for them.