I was able to play a little more with this experiment, and any folder within C:\Sandbox\Sandbox_Folder\* will still get permissions set by Sandboxie itself. There just isn't a way that I know of to prevent this from happening.
To clarify and make it more understandable for others not understanding much of what I have been mentioning thus far, this is what I managed to change so far:
C:\Sandbox - I removed the default permissions, and applied the permissions of C:\Users to it. They remain.
C:\Sandbox\Sandbox_Folder - I removed the default permissions, and applied the permissions of C:\Users\User_Profile. They remain, provided that I don't delete contents with Sandboxie. I use Eraser to delete the contents of the folder, but not the folder itself.
Any other container/object (folder, file...) created within C:\Sandbox\Sandbox_Folder have permissions set by Sandboxie itself, and they cannot be changed, at all. These permissions are for the group Authenticated Users and the User.account.
It would be nice if there is a way to force containers/objects within C:\Sandbox\Sandbox_Folder\*
to inherit from the parent folder (C:\Sandbox\Sandbox_Folder).
The only way I'm thinking, at the moment, and that could possibly do the trick, would be to script something that detects the creating of any new child objects and automatically recreate the permissions from C:\Sandbox\Sandbox_Folder to them.