With QGIS Lisboa, there is now a basic support for Goode Homolosine projection.
I downloaded IGH data from http://edc2.usgs.gov/glcc/fao/data_specifications.php
and set up this user CRS:
+proj=goode +lon_0=-100 +x_0=-11119487.43 +y_0=0 +a=6370997 +b=6370997 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
for North America.
lon_0 and x_0 are from Appendix 03 of ...
There is a bug in the Project Raster Tool. The output raster from the Project Raster tool is a little too big on the eastern edge, and on the southern edge (bottom of the easternmost gore/lobe). So the data extent doesn't quite match what the extent of a Goode's Homolosine projection is, and triggers the "not consistent extent" warning.
A workaround is to ...
Yes, Earth Engine can reproject images to non-EPSG coordinate reference systems.
Earth Engine uses the Java GeoTools library for coordinate system transformations which supports the Coordinate System Well-Known Text (WKT) format, and a set of base projections.
The ESRI:54009 World Mollweide coordinate reference system (CRS) has a WKT ...
Neither +proj=igh or +proj=goode are currently supported by proj4.js but mproj does support +proj=goode Demo here https://jsfiddle.net/3bjo7rsL/ and the code below. Error catching is needed to handle "off the world" transforms at the edges.
I would have a try by creating a fake EPSG code into the lookup table "epsg" that is located under the proj directory. On Linux it may be found as share/proj/epsg, in MS4W installation on Windows as ms4w\proj\nad\epsg.
The epsg file has records like
# WGS 84
<4326> +proj=longlat +datum=WGS84 +no_defs <>
Add your own code as
# Goode's ...
The data specifications page says that the earth model aka geographic coordinate system aka datum is based on a sphere of 6370997.0 m. The Goode Homolosine (Land) definition in ArcGIS is based on WGS 1984. When I modify the definition to use Sphere (ArcInfo), which matches the data specification information, I can overlay country boundaries quite well.
QGIS 2.18 is nigh on fully supporting the Homolosine for rasters. You just need to be aware of issue #959: perform any raster warpping directly with GDAL using the CHECK_WITH_INVERT_PROJ parameters and you are good.
Regarding vectors things are still complex. Some of the errors AndreJ shows in his answer prevail. The cause is simple: QGis is not aware that ...
There has been a question on interrupted projections before: Discontinuous or Interrupted map projections?
The conclusion was that proj4 (which is used for projection handling in QGIS and GRASS) does not support interrupted projections.