I remember I have done something like that to mark contour lines defined by the distance from a coast. First I map the land onto a sphere in 3D, where I can use EuclideanDistance. From that I define a function of latitude-longitude and make a ContourPlot. Finally I map the contours into GeoGraphics inserting GeoPosition between Line[ and the list of points.

(1) is exactly what I need. Thank you! I should have come up with that myself.

(2) is more complicated, I think. You have a single disk centered on the position associated with NYC, (presumably a centroid, I imagine). What I want would be the union of all the disks centered on the points on the city limits, i.e., the points you collected in "points" in (1), and all the points along the segments joining those points. It would be easy to write a mathematical description of this set, but I'm not sure how to get it in a form where GeoGraphics can use it.

For (1)

polygon = Entity["City", {"NewYork", "NewYork", "UnitedStates"}]["Polygon"]
points = List @@ First@polygon

For (2), do you mean something like this?

nycPosition = Entity["City", {"NewYork", "NewYork", "UnitedStates"}]["Position"]
GeoGraphics[GeoDisk[nycPosition, Quantity[50, "Miles"]]]