The Apostle St. Thomas (AKA doubting Thomas) is credited with introducing Christianity to India. Many of the Orthodox churches in India are known as St. Thomas Apostle Indian Orthodox Churches. The funny thing is that, even as professed Christians, these people still separate themselves into castes. There are separate churches for the higher class and those considered lower class. I guess old habits die hard.
Yes, in Kerala and Goa, they do. However, this is not a carry-over from Hinduism. The discrimination here is between the earliest converts a la St.Thomas, and the latter day ones, whom the former consider inferior, since they weren't converted by an apostle of Christ.
Inculturation at work, badly. Supposedly Ghandi turned away from Christianity because its Indian incarnation preserved the caste system.