When these rasters are combined using the "Combine" tool in the spatial analyst tool box (not considering arcpy), what method is used to determine the value in the new combined 3x3 raster? '
This is a really old question, but it popped up and deserves an answer in the thread.
Depending on the environment settings, the processing will happen at the cell size you specify in the environment. Resampling on the fly like this is done using nearest-neighbor interpolation.
Here are some examples based on the cellSize environment setting:
If you want to do any other kind of sampling, this is done before the Combine using other tools (for example Resample or Aggregate).