Is there a way to drop in this widget (currently a separate page with html, css, and javascript) into my webpage without the webpage applying all of its own CSS on top?
The easiest way would be to use an iframe
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If you want an embedded widget, the wisest thing to do is to specify the CSS properties inline in the widget's markup. It's bad practice usually, but is the right thing here.
Check out for example what the Facebook Profile badge (you need to be a Facebook user to see it) looks like:
<!-- Facebook Badge START --><a href="http://www.facebook.com/pekkagaiser" target="_TOP" style="font-family: "lucida grande",tahoma,verdana,arial,sans-serif; font-size: 11px; font-variant: normal; font-style: normal; font-weight: normal; color: #3B5998; text-decoration: none;" title="Pekka Gaiser">Pekka Gaiser</a><br/><a href="http://www.facebook.com/pekkagaiser" target="_TOP" title="Pekka Gaiser"><img src="http://badge.facebook.com/badge/1238473725.2447.1697336437.png" width="120" height="239" style="border: 0px;" /></a><br/><a href="http://www.facebook.com/badges/" target="_TOP" style="font-family: "lucida grande",tahoma,verdana,arial,sans-serif; font-size: 11px; font-variant: normal; font-style: normal; font-weight: normal; color: #3B5998; text-decoration: none;" title="Make your own badge!">Create Your Badge</a><!-- Facebook Badge END -->
They are very likely to have done a lot of testing on this, so the CSS properties they set in their code, I would recommend to set in your widget, too.