Does Mozilla ever scan a computer and send a message that there are viruss on it and prompts me to download a file to get rid of them?
I have also gotten messages saying that Mozzila has detected unusual activity from my network and asks me to type in a phrase to continue. I am running AVG 2011 and it has not detected any viruses
Opaite Mbohovái (1)
No, Mozilla does not scan your computer for viruses. That most likely is a ploy to get you to install malware.
Regarding typing in a phrase to continue, occasionally I see that when searching Google very repetitively. Google will show a page asking you to prove that you are human. It wants to discourage automated access (bots). But that comes up in the Google search page, it isn't a popup or dialog and doesn't mention viruses.