If you just list several quotes from one of their pages on one of your pages it could cause problems. But if you just do one short quote per page and then do an analysis of what the quote means or if you had an article in which the quote is just a small part there shouldn't be a problem. That way the quote from another webpage is just a small part of the overall page.
Be sure to set up a reference of some sort to the original source. I don't know if he cite tags help but they wouldn't hurt. I'm not sure if <cite> is that meaningful anymore.
nav bars aren't repeated content, all websites have navbars. People are getting really carried away here with very little empirical reasoning.
The only real duplicate content issue you would have is if you simply copy and pasted in a gov web page into your webpage. Any other use is fine. Any normal, fair use type quoting is just one block of some text within other blocks of text.
Google only pays the barest of attention to tags, and I would give you 100 to 1 odds the cite tag is totally ignored. Like most other tags, by the way.