1) $ARCHOME is a environment variable for ArcInfo Workstation. If you are using a Windows machine, you can find what path this variable points to in many ways. But probably the easiest is to:
a. Go to your desktop, and right click on My Computer > Properties > Advanced (tab) > Environment Variables.
b. Look under the System Variables, and you should see one for ARCHOME (probably something like C:\Program Files\....
2) Okay, I get it. Question: Did your original .tif files have accompanying .tfw files?
If so, all you need to do is copy those into the location where your new photoshopped .tif files are, and make sure the have the same prefix name (for example: photoshop_test4.tfw goes with photoshop_test4.tif).
If your originals .tif files (the ones you used as input to your Photoshop process) did not have .tfws (thus indicating they were either geotiffs - or were not projected in the 1st place?!?), it will be a bit harder, but still pretty easy. Take a look at the ArcGIS tool called "
Export Raster World File (Data Management)". Basically you would have to write a script that would look through all your original .tif files, run the tool on them, and then write the output .tfw files to the path of your new post-photoshopped images. There are harder ways to do this as well, but let me know if either of these two methods works.