How to convert .json files to .jpeg

Brad, I thought I had made it clear that none of your photos are being changed in any way when you backup your catalog.  The .json files are files that are related to the database and face recognition.  There is one .json file for every photo that has been face recognized during media analysis of your catalog.  Unfortunately, those files are very small and create an exponential addition of time to creating and restoring the backup.  But if you were to delete them all, you would lose any face recognition information that has been created in your catalog.

I'm not sure what you mean by a photo stick.  If you are talking about a USB thumb drive, the answer depends on the size of the device.  I have seen some as large as several Terabytes, but the ones I tried were unreliable.  Most are much smaller e.g. 256 GB or much less.  You would need to make sure that there was enough room for all of your photo files as well as the catalog folder files which will be relatively small, depending on the size of the catalog.

A better approach would be to use an external drive.  Huge external drives are now very affordable.

Probably the best way to transfer the photos and media catalog from your old to your new computers is to try to create a backup of your catalog on the old computer and restore it to the new one.  Once the catalog and media have been transferred to the new computer, you can convert the catalog in Lightroom via the File menu:

Before creating a new backup on your old computer, use the File>Manage Catalogs dialog to Repair and Optimize the catalog.  Also, make sure that you have no missing files in the catalog, by using the Find>All Missing Files command.

You can be assured that your jpeg files were not converted to .json files which are text files associated with your photos.  What you are seeing is probably the files that contain all of the face recognition information in your catalog.  How did you come the conclusion that your photo files have disappeared during the backup process?

And a catalog backup is not intended to simply save your photos to an external drive.  The intention is that you can restore the backup to replace your original photos in case they are lost from their original storage location, e.g. if your hard drive fails. 

So, what OS and version of Elements are you using?