Using an iPhone 5 (Verizon version) in Europe and India?
Yes.
Yes.
Yes.


Only quirk is you won’t get LTE data on networks other than Verizon, especially Airtel and Aircel, as Asian carriers use different bands. You are probably gonna get LTE in Europe, and that too most likely on CDMA networks, but you will have to check with the carrier. I really hate this fragmentation.