Copper will still be used, its just the traditional analogue phone signal delivered over the copper that is stopping. New (and changed/upgraded) users will instead receive a digital phone service, effectively over broadband (which can still be delivered over copper lines).
All checking is done locally on your machine from a hashed list of "bad" domains, your visits aren't sent to google. You can get the full details here: https://support.mozilla.org/en-US/kb/how-does-phishing-and-malware-protection-work