Copying text from a website has no effect on your system fonts, regardless of whether you save what is copied. Furthermore, it is unlikely any font used by a web page isn't supported by Word. What is more likely is that it might use a font that's not installed on your system, which is an entirely different issue. In such cases, Word will substitute another font. This behaviour is quite normal. Although some printers come with their own fonts, which supplement the ones Windows and Office come with, that too has little or no bearing on the matter.

On the whole, your description suggests you either have a faulty Office installation or some corrupt fonts. Have you tried repairing the Office installation (via Start > Windows Control Panel > Programs > Programs & Features > Microsoft Office (version) > Change > Repair) or reinstalling your fonts?

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Cheers,
Paul Edstein
[Fmr MS MVP - Word]