Do Apple products bought in US etc have proper warranty and Apple care benefits in India?

Yes. Source: me. I had asked them. This was a couple of years back. I've gotten my whole screen replaced in india under warranty for a MBP bought in the US. You can ask them if you want. The online chat support is quite good.

Yup, any apple product bought in US is covered fully in India (including Apple Care+ benefits).

Great thanks!