I would like to use some projections in QGIS that I can't see supported directly. An example would be something like this conformal cube projection. http://www.csiss.org/map-projections/Polyhedral_Globes/Cube_Gnomonic.pdf Any hints on finding details and using this or similar projections?
QGIS depends on GDAL, which is build upon PROJ.4. So you are out of luck in QGIS if there is no PROJ.4 implementation of the projection you need.
You can check
proj -lp in the OSGEO4W shell or a linux terminal to see which projections are covered.
Unfortunately, GDAL only allows projections that come with the implemented inverse projection as well. This leaves only 48 out of 134 PROJ.4 projections.
For raster files in other projections, it might work to calculate X and Y coordinates of every cell programmatically, and store them in separate bands of a VRT file. Similar to the way MODIS Swath and other HDF files are processed, you could run
gdalwarp -geoloc on that. But that would not work with vector data.