Well, first of all, I don't know about viruses or trojans, but when internet is slow the first symptom is always spyware. Use a free spyware scanner such as Ad-Aware and Spybot, then be "quite sure" (lol) if you have no viruses.
As for the internet randomly dying, you can always ask your ISP and make sure first that it's not them! But try these guys' ideas first, their usually right.