I was wondering if a hologram could be made that could focus light that wasn't in the original hologram. I'm assuming that this would have to be a mathematically constructed hologram as, by my understanding, a hologram of an actual lens can only focus light of images in the actual hologram.
The hologram of a single coherent point source is called a zone plate and acts as a lens. You can understand this by thinking about the typical operation of the hologram: it takes a reference beam of parallel light rays in and puts out a set of rays that converge at the desired point. But that's just what a lens does. The zone plate is more sensitive to changes in the wavelength and direction of the incoming light than a typical refractive lens; see the linked page for the distinctive effect on photos taken with a zone plate.
A home printer these days often has just enough resolution to directly print out such a zone plate, if the pattern is computed beforehand. The "traditional method" is purportedly drawing out a large version of the plate and taking a photo of it on film to get it to the right size. If you tried to holographically record one, you'd need a coherent point source, and a simple way to get one is to pass the hologram's illumination laser through a lens. So in some sense the answer to the question "which holograms act like lenses?" is simply "holograms of lenses." The caveat is that that means holograms of strictly lenses; the effect is lost if the illuminating beam hits a scatterer before it hits the lens, which is the idea I think you were getting at. Conceivably you could mix the effects if you took a hologram of a scene where a lens is partly illuminated directly by the laser and partly by a scatterer reflecting the laser. Illuminating the resulting hologram with something other the reference beam would superpose the result of passing the illumination through the lens (desired) with a blur of light from the failed reconstruction of the scatterer (undesired, but maybe controllable).
Yes, indeed! I’m not sure whether you could make such a hologram the old-fashioned way, but nowadays you can engineer a flat lens by varying the phase profile appropriately across a surface. Look up “metasurface lens”. These are typically made of nanofabricated structures on a transparent substrate, or I suppose you could use a spatial light modulator.