No, you can find out with optical test equipment such as an OTDR, but not with the less-sophisticated telemetry available in a router/switch.
You can know the path loss (limited by the accuracy of the optics' Tx/Rx power meters) but that loss includes things like bends, dirt, connectors, etc. so it's not usable as a good estimate of distance.
SONET, or rather the synchronous digital hierarchy it's running on, requires very tight clock coupling across the whole structure. In order to stay within the sub-millisecond tolerances, there's no other way than with atomic clocks.
Sharing a central clock wouldn't work due to the inherent limitation of the speed of light, which covers just 300 km in one ms (...
The transformer is there mainly to decouple DC common mode signals, as has been explained in other answers, and it also provides the electrical isolation needed in practical data networks (as opposed to running the entire network on your bench, all powered from the same extension strip, for example). The transformer manufacturer will state which IEEE 802.3 ...
SONET defines a physical layer (or variants thereof) and a common data link layer, similar to Ethernet.
(Circuit-switching) SONET used to be a popular carrier technology but has largely been replaced by (packet-switching, much more efficient) Ethernet.