Going from the court decisions, they would know that traffic is illegal if a court tells them it's illegal. Again, refer to People v. Brophy: the common carrier is not "authorized or required" to investigate. Probably meaning they can aid in investigation (e.g., supply information to police) but not perform an investigation (analyze said information themselves).
In addition, Title II would probably put internet traffic on the same level as wiretaps in terms of (non-DHS) data snooping. If anything, it would give internet users more protection for this sort of thing than they are granted now.
It would probably be a specific url. Or an entire domain for something like a site dedicated to illegal gambling. Though why they wouldn't just go after the website to take it down like they do currently eludes me.
Pretty much. If it's hosted in the US, they'll take it down. If it's not, they'll do pretty much whatever they can ... which boils down to seizing the domain. Usually with the kinds of sites people are typically worried about USGOV taking down, they leave up ... and watch the traffic. It's like in The Wire - they don't want the criminal's site (cell phone) to go down when they're watching the connections it's making.
Title II combined with the 2nd amendment actually prevents things like The Great Firewall :)
In addition, Title II would probably put internet traffic on the same level as wiretaps in terms of (non-DHS) data snooping. If anything, it would give internet users more protection for this sort of thing than they are granted now.