Hi boho,
Your reinstallation on the desktop PC shouldn't have affected installation on the laptop and, if it did, you'd simply need to call MS to get the installation activated - nothing would prevent the actual installation, though.
I suspect your problem can be traced back to the Office 2010 trial. You needed to completely remove that before attempting to install Office 2007.
Since you might now need to completely remove both Office 2010 and Office 2007 before you can do a fresh install, see:
http://support.microsoft.com/kb/290301