Hello again, we already replied to your support ticket today.
The license allows you to install one copy of the software on one computer. Transfer from one machine to another is only allowed when this is absolutely necessary (e.g. after reinstalling Windows or when you buy a new computer). Anyway, you can uninstall the software from your laptop now and then activate it on your desktop again. Nothing should happen to your project which is stored on your desktop.
As to the last issue: you probably could not play the slideshow on your laptop because the paths to image files changed.
Anyway, you don't need to install the software and transfer the project to your laptop. Just create a video file on your desktop and take it to your laptop to play.
I am still not sure whether you played the slideshow project right from the software or converted it to video first.
If it was the video, which format did you choose? (AVI, MP4, HD-ready, full HD etc.)
Did you try playing the video with Windows Media Player, Media Player Classic, or some other software?
And one more question: can you play the slideshow (the project and the resulting video) on your desktop with no issues, does it look fine there?