Though council tax rarely aligns with desirability or average property price of a borough. Funny you picked Westminster for the example, the one with the cheapest council tax of all.
Tell me about it. Currently paying way more council tax on a way crappier flat in Tower Hamlets (one of London's poorest boroughs) than we used to pay on a beautiful house in Hammersmith and Fulham (one of London's richest...)
Yes, it's basically a residential property tax as it's based (very loosely) on property value and is payed per-property, not per-resident. But it falls on the resident, not the property owner.
Called council tax because it funds council services like street maintenance and rubbish collection.
I don't think there is a property tax in the UK - not as a yearly recurring payment that (some of?) the US has. There's Stamp Duty though, which is paid when you buy a property