Can I cut through hollow-core closet doors (sliders) with a utility
knife? I
would like to put pulls in them, and I don't really have a way to cut
the
holes any other way.

Sure can, take a look at the side profile of the door, most likely it
will be a solid center sandwiched between thin ply or MDF, even if it
has a wood veneer. It will give you an idea of how thick the exterior
is. I recently added a window in a pocket door and cut the hole with a
new bladed utility knife...worked well. I was cutting a rectangle,
circles beg for a hole saw of the appropriate diameter.

Good luck,

Darwin