Specifically, I’m asking how the glands deliver the sweat to the skin surface. Do they push the sweat out, or is there something on the surface that pulls it out? How do antiperspirants work?
No sweat.
The gland consists of a cluster of cells at the base and a tube which empties at the skin surface. Sweat is produced at the base of the gland and is pushed to the surface by the sweat which follows it. There may be a small component of wicking of the fluid upwards (ie pulling), but the driving force is the hydrostatic pressure created by sweat formation.
Here is an micrograph of a human skin section. The base of a sweat gland is in the box on the lower right of the image. For orientation, the skin surface is at the left.
As for question #2, they work by physically plugging the sweat glands. The results of this search should provide you with more than enough info about sweaty pits and anti-perspirants.