According to spatilreference.org, your proj4 string is not correct. -a_srs defines the projection but does not modify the coordinates, so your output vrt file is likely to be incorrect (based on a sphere and not an ellipsoid, longitude of origin is 100 and intersection at parallel -70 instead of -60). Below is the proj4 string for EPSG:3412
+proj=stere +lat_0=-90 +lat_ts=-70 +lon_0=0 +k=1 +x_0=0 +y_0=0 +a=6378273 +b=6356889.449 +units=m +no_defs
If you are sure that your input data is in NSIDC Sea Ice Polar Stereographic South, you don't need the first line of code and you could write :
gdalwarp -overwrite -s_srs EPSG:3412 -t_srs EPSG:4326 -te -180 -90 180 90 input.tif projected.tif
EDIT: 1) in the proj4 string, you can read the parameters of the projection:
lat_0 is the latitude of origin, that is the summit of the cone in case of stereographic projection
lat_ts is the latitude of the parallel where the cone intersects the ellipsoid
lon_0 is the longitude of the origin of the projection
k is a scale factor
x_0 and y_0 are the shifts of the origin of the cartesian coordinate system (it is used to avoid negative coordinates with some projection)
a and b are the size (in m) of the semi-major axes of the ellipsoid (when they are the same, you have a sphere)
2) Your output pixel size will be in degrees and you can set it using -tr. An average pixel size could be your original pixel size in meter divided by 110000 (this is roughly the size in meter of one degree along a meridian). However, stereographic projection and WGS84 coordinates are very different. Therefore there will therefore be a large amount of resampling due to distortions, and you might have to optimize your pixel size for the location where your work is focused.
