First of all, I would recommend you question if there is a need to have the imagery at the same resolution and pixel aligned. Such a process will always result in datasets being resampled. Resampling imagery by definition will decrease the quality of the imagery as you will have to interpolate a new pixel value from neighboring pixels, else get nearest neighbor sampling artifacts. One of the key capabilities of ArcGIS is that it performs such resampling on the fly. This enables the data to remain in their purest original form and only a single sampling is performed from the source to the pixels on screen, or the defined output extents and pixel size of any analysis that you perform. This handles all sorts of other changes that might occur such as reprojection with only a single controllable sampling of the pixels occuring. The additional advantages of this are that there is no additional disk space requirements and one can always refine parameters such as the georeferencing without affecting quality. If you need multiple images to be available as a single rasters you can use a mosaic dataset or if just working with a few images use the mosaic function that is part of the IAW (Image Analysis Window).
Once you have a raster layer created from one or more rasters (or a mosaic dataset), if you do want to create a new raster then there are a number of options. You can either use the ‘Data/Export Data’ tool from the ‘Table of content’ and define all the parameters such as the extents and the number of rows/cols etc. The tool enables all parameters that affect the extent to be defined. Note that the resampling method used is that defined as part of the layer properties. Alternatively use Copy Raster tool, this honors the Environment variables including Extent and Snap. Again you can ensure that the output raster is pixel aligned. There are tools such as split raster that have more options on how to output the data. (Note in most cases I would recommend exporting imagery as TIF)
To the question of how to update the georeferencing. Just use the georeferencing tool but use the ‘Update Georeferencing’ option. This just adds geometric transforms to the raster without changing any of the pixel values (no sampling). If you do a ‘Rectify’, this is similar to doing an Export which results in the data being resampled, but the rectify tool does not honor the environment variables. This is why I would suggest one of the above methods to persist the image.
In answer to the question of order: Update georeferencing and check that the output is as required. Just use, these should be few reasons to need to export. If you do need to export use on of the methods above. Remember that best results will be obtained by not unnecessarily resampling your imagery.
You will find that ArcGIS is a very ‘robust’ tool for performing all such image processing and analysis. Do rethink workflows so to remove the intermediate outputs that are typical with many of the other image processing tools.
