All of the physical images are in a named folder in the pictures folder on my internal hard drive, e.g., C:\Users\username\Pictures\folder\folder\folder etc. So, when I move to an external hard drive, do I have to use the same long folder structure or can I change it?
You can change it but if you do it outside of LR then you will need to Re-Link the LR catalog file to the new names of the folders the images are in.
If you do it inside of LR then the LR catalog will be auto updated, of course.
If you really want to change the folder and file structure/names:
Copy the Pictures folder to the new hard drive and link LR to those folders as shown in the above posts.
Then, WITHIN LR once you have re-linked the folders, you can rename the folders/files to your hearts content by right clicking. Be sure to back up the new drive and the LR catalog.