My phone was purchased the US, and don't have purchase proof. Can I get it swapped in India?
fbara is correct - there are too many variables for anyone here to give you a definite answer here. But here is what I can tell you. 

If your iPhone is still covered by Apple's global warranty, then it will still be covered. If you purchased an Applecare plan, and it's still within the period of cover, then it will be covered. If it's out of Apple's own warranty period, it may still be covered depending on your local (India's) consumer laws.

Whether or not Apple will swap your phone will be up to them to determine. They will need to assess the faults and will then decide whether it's a repair or replacement proposition, or indeed one in which you need to pay some money.

One thing you can do is check your serial number here. 

Checking your serial number will advise you if it still has a valid purchase date, what repairs and service coverage it has, and whether you're still eligible for technical support over the phone. If you're not sure where to find your Serial number, the easiest way to do it is to go to Settings > General > About and it's listed about halfway down.

Finally, it is definitely still worth you calling Apple about this - even if checking your serial number shows your coverage has ended. Apple has been known to still honour repairs and replacements for products outside warranty periods depending on individual circumstances. They may also be able to arrange a swap by putting a hold on your credit card to the value of a replacement phone and just shipping you a replacement. As long as you send the other one back within a certain period, you'll never be charged.

In summary, just call Apple. In India the toll free number is 000800 1009009 (as long as you're calling from a landline, otherwise you may be charged by your provider if calling from a mobile).