Personally I'd create a mosaic dataset (or VRT) and extract the area of interest from that.
VRT is supported by Global Mapper but will be quite slow, I would go with the mosaic dataset for this operation.
Both of these raster types are links only to their respective rasters so don't take very long to create as the rasters themselves aren't being copied. Mosaic datasets will create overviews and will take a bit longer - but that's probably worth it.
Extracting from ArcGis can be done in a few ways:
To get a list of all the rasters in all subfolders you can use
Dir in DOS like this:
DIR c:\where\images\are\*.tif /b /s > c:\some\path\TiffFileList.txt
Which instructs command to find all the files with .tif extension in this folder and all subfolders and write just their full path to a file c:\some\path\TiffFileList.txt - obviously substitute your own paths and extension if not tif. If your paths contain spaces they will need to be quoted like
DIR "c:\path with spaces\*.tif" /B/S > c:\some\path\TiffFileList.txt or the instruction will get confused.
Then open the text file, copy the contents and paste into load rasters for a mosaic dataset or use
-input_file_list c:\some\path\TiffFileList.txt for GDALBuildVRT.
If you know which rasters you want to copy you can open up the file list in Excel, delete the rows that you're not interested in, insert a column and fill with
copy then save as csv - open the csv in notepad and replace the commas with spaces then save as .bat and you've got a batch file to copy rasters... put the batch file in the location you're wanting to copy the files to and double click. You might run into problems if you have duplicate file names or spaces in your paths though.