What Intersection algorithm do you use? In case you perform a triangle intersection check, you're likely to receive the barycentric coordinates of the point on the triangle. Use them to compute the point of intersection and get
your collision normal with "collisionNormal = Sphere.Center - PointOfIntersection"... You might want to normalize it though.
That's all I can say with what you've posted so far.

Good luck!

Martin

<edit> Do you want to take movement into account?</edit>

The collision normal of a sphere with anything is always along a line from the centre of the sphere to the contact point - along a radius.

As for whether it collides with a corner, edge or face this is determined by the location of the sphere's centre. One way of looking at it is as follows:

Suppose the box is defined by

x1 < x < x2
y1 < y < y2
z1 < z < z2

and the sphere is at (X, Y, Z)

Simply count how many of the following are true

x1 < X < x2
y1 < Y < y2
z1 < Z < z2


If none are true the sphere hits a corner. If one is true the sphere hits an edge. If two are true the sphere hits a face (if three are true the sphere is inside the box, which is case you probably want to exclude).

More formally the planes x = x1, x = x2 etc. divide the space into Voronoi regions, of which this is an application. You can work them out for other polygonal bodies, though the maths quickly gets a lot more complicated.