You think if I take my phone to lamington road, Mumbai, will they change it?
not everyone here is aware of that place you mention, but if you are hinting towards an unauthorized person digging up your device for replacement i personally dont practice or promote it unless i have decided to buy a new device and just drag the old one for a couple of days or weeks.
to answer your question, yes they should change but the reliability and the quality cannot be guaranteed which you expect form authorized centers.
no clue about that but as you already have knowledge on the price over the internet you can expect something in that range, but if you choose the genuine once be ready to pay the full price.
i will say yes you should , lumia 1020 is already 3 years old and is a vegetable in terms of support and app updates. With apps being pulled or being abandoned form windows phone 8.1 store , this is the best time for you to look for other options in the market.