I've just bought a new laptop and I'd like to use RWIE on it and uninstall it from my old laptop, but I can't find a place for entering an existing licence code, I can only find links for buying a new code. Where do I go to activate the evaluation copy with my existing licence code?

You can access the registration window from the About box (Help->About RealWorld Icon Editor). Or you can wait 25 days and it will appear automatically when starting the application.