Downloading Photoshop Elements 15 onto a new computer

I recently purchased Photoshop Elements 15 and downloaded it from the Adobe website.  At the time I was told I could download it again when I changed computers.  I now need to replace my old laptop and the computer shop will copy all my documents etc onto the new computer.  I'm trying to find out if the shop needs to download the software in order to transfer the images I edited on Photoshop, or whether they can move the images to the new computer then I download the software before being able to open them.  I also want to check whether my registration key code will still activate the software.  If not, how do I get a new key code?

Thanks for any help with this.