Off the top of my head (check this yourself):

Given a point c, tip of the cone position p, and cone direction d:

Setup vector C = c-p

Length of C = |C| is distance from c to p.

Let Ldot = d dot C

[EDIT: a little bit of a slipup; the following assumes d is the normalized (length=1) direction vector]

if Ldot < 0, point c is not in the cone (it's somewhere "above" the cone in the opposite direction of the cone direction)

if Ldot > range, point c is not in the cone. (it's beyond the end of the cone)

Since Ldot = |d| |C| cos(angle),

Let angle = cos-1( Ldot/( |d| * |C| ) )

if angle > cone angle, point c is not in the cone. (angle is what I think you mean by "field of view.")

It is possible to test a cone for intersection with a plane, and the algorithm is described in the book Real Time Collision Detection. If you perform this test against the 6 planes of the view frustum, you can cull spotlight cones in the same way that you would typically cull a sphere. Just be aware that as the cone gets larger relative to the size of the frustum it becomes more likely to generate false positives (the same problem happens with spheres as well). As an alternative you can represent your spotlight with an oriented bounding box or a frustum, and test for intersection with a frustum using the separating axis test. SAT produces exact results, but is more complex than testing a sphere or cone against the frustum planes.

As for scaling down your shadow map resolution, you should consider that in the ideal case you'll want to scale your shadow resolution based on the projection of your occluders onto the receivers being lit by the spotlight. So if you're going to base it on a distance from the camera, then you'll probably want to pick a point inside the cone that represents the most likely place for a receiver to be. If everything is dynamic then you'll probably have to guess or estimate, unless you do some calculations at runtime. However if the light is static and there's static geometry involved, then you can possibly do a bit of offline analysis to make a better initial guess at which point to use.