I have felt this pain before! What you need to do is get a JPEG, any old image, and create a world file with these parameters:
Use notepad to create the file and then rename
.jgw, the base file name should be exactly the same as the jpeg, for example if the image is Family.jpg the world is Family.jgw. Now it's falsely georeferenced. In your environment settings set this as Snap Raster so if you do something like polygon to raster its cells will align with that georeference.
Now, set your output extent to start at 0,0 and extend to cover all (and some) of your inputs. Now when geoprocessing rasters the cells will align with and start at 0,0.
Be aware though that some tools will shrink the extent if it is all NoData to the left and will not actually start at 0,0; to get around this you can try put in a fake feature at -1,-1 which will be trimmed off by the extent. Even then you may experience shrinkage if there's all NoData.
Another option is to make the background a value and then set the NoData value via catalog after processing. This all depends on which tool(s) you are attempting to use.
To create a fresh (empty) raster you can use Create Raster Dataset with these settings, though width and height are not implicitly set in the tool it should get enough cells to cover the Extent environment. To create a raster filled with a background value create a polygon feature class, make it the same spatial reference as your data, edit it in ArcMap and draw a box polygon big enough (absolute x,y might help here) and then use Polygon to Raster to convert it into a raster dataset.