Depends on the service centre - AFAIK, mobile phone ASSC demand VAT bill. Sony does the same for their laptops - and no bill means you are out of luck in some cases.
Not impossible. I know of one store here in Delhi which sold me an imported product once. This was verified by the company customer care online through the S/N. It was supposedly imported from and to have been sold in Singapore only. I've never went back to that store since.
If its imported and dont have global warranty they wont accept the product. Legally purchased goods and goods for which taxes are paid for are registered before shipping it to the market. This is why service centres ask for S/N when you dont have proper invoice. Apparently S/N is registered with the importer/service centres immediately after importing. I saw this personally once in a distributor's office.