They don't work the same way they used to though. Since old school PSTN hardware is being replaced with VoIP, there is a hacky protocol called T.38 which does just enough to convince each side that it's talking to a real fax, and decodes and forwards the data over IP.
T.38 is actually a fine way of transporting fax bits but unfortunately it is quite uncommon to see T.38 end-to-end. While a VoIP provider may negotiate T.38 with a customer's fax ATA, it is likely being transcoded to G.711 by a gateway at some point as it traverses the telephone network, ultimately making T.38 a less-reliable choice. (Better to have the same codec end-to-end.) A comparison might be cellular or VoIP providers offering wideband codecs, which sound great when you stay on-network, but when the call crosses the PSTN and is transcoded, the sound is worse than if you used the standard narrowband end-to-end.
There's nothing wrong with the wire protocol, but T.38 was designed to work with faxes that don't know it exists. Given that VoIP doesn't provide the same highly-constant latency as the PSTN, T.38 gateways pull some dodgy stuff, such as (if I recall correctly - I no longer have access to this code) deliberately introducing HDLC CRC errors to give themselves time to wait for a packet to appear.
Circuit switched class five offices are still very plentiful though, and DS3-based transit networks are still nationwide. So if you want it, you can absolutely still experience phone networks without voip.
Depends where you are. In the UK, decommissioning of the PSTN has started and will be completed in the last few places by the end of next year. There will still be a "phone service" for those that want an equivalent service to their old landline, but it will be provided by a VoIP box.
I doubt that very much. Pretty much no telco has used circuit switching in decades. Everyone moved to packet switched network a long long time ago. Even if you have a Real Landline, it's just plugged into a VoIP box at your nearest telco branch.
Uh, no. As someone that's worked on these switches (and transport gear associated with them), you can still go from one end of the United States to the other without hitting a packet switch. There's still thousands of them out there.
I'm not sure if you can get native DSx/SONET backhaul in Canada from one end of the country to the next, but I know it's a similar situation there with end offices. Bell also operates a pretty extensive network of DMS-250s, so a lot of telecommunications traffic hits circuit switches regardless of its destination.
