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...
That was the old requirement, but none such exists now. Anywhere can be granted city status these days, cathedral or not.
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
a city, without a cathedral its not a city