As long as you are connected to the internet the jquery stuff should work without hosting the main jquery files locally. If you are working ith a laptop and not always connected to the internet then the link about private networks may be relevant.

If you include a stylesheet with css for styling a jquery widget it should override any css included from google because your css will be included below the css from google and this should override styles from the google jquery css.

I would use Firebug plugin to inspect the html of the jquery widget after the page renders and make sure your css is using correct classes to style it. You can also inspect the network traffic with Firebug and see if it is getting any 404 errors in pulling in images declared in your css.

Hope it helps,

Joe