You are asking to see meticulous proofs in a simple blog post, which is not gonna happen. Also, any article like this will assume that you already know something about what's going on there, or at least in the rest of the world. Eg: If someone says 20-30% inflation rate is high, it's obviously comparing it to the inflation in other countries. So yes, it's damn high.
About the metrics, its pretty simple: they just pick the products/services that didn't increase much their price during that period. Of course nobody can't actually live with those products, nobody buys them or even know them. And when they need something real for the index, like the Big Mac.. Well, you know what happens. Automagically its price is reduced to a half, and the product is removed from the menu.