No, cities can have cathedrals (although not necessarily Catholic) to be cities but other things like good universities (e.g. Cambridge) can also allow a city to be designated as such. Factors like...
No, a city is a town over a certain size religion does not come into it.
There is a city of this name in Riverside County, California
That was the old requirement, but none such exists now. Anywhere can be granted city status these days, cathedral or not.
Both which and what can be used