need to find a web site where I type in the name of a city anywhere in
the world and get the names of all airlines which fly in and out of
that city.

I don't know of any site like that, but here's a site that provides
links to airports almost everywhere. Often an airport's web site will
include a list of the airlines that serve it.
http://www.airlineandairportlinks.com/

This site may help. It gives the abbreviated code for all airports and
will give a list of all flights
http://www.oag.com/