Creating polygons over maps and extract the latitude/longitude?

Both the Bing Maps and Google Maps APIs could be coded to do the first (draw polygon and fetch vertices).

Points in polygons are more difficult if you are attached to a web page interface. If you are happy to put the polygons in a proper database, then Postgres/PostGIS, SQL Server, and Oracle can all do proper spherical point-in-polygon tests.

If the polygon is small (so that a Euclidean approximation is acceptable), then it can be easily coded up yourself. Point-in-polygon on a sphere is more involved - I've coded it up in C# in the past (been there, done that...)

Finally, querying cities: The Geonames database ( geonames.org ) is useful. You could easily filter the data to extract the city coordinates (ie. discard other features, POIs etc), and store those in a data file to query or in the database.