Electronic – the function of this choke-like transformer on the PCB side of this RJ45 jack+magnetics connector

Ethernet magnetics do vary, in particular in turns ratio between the network and circuit sides, and other parameters like center taps and baluns. The ethernet signalling voltages are well defined, but some PHYs operate at different voltages. Since ethernet is transformer coupled, that's no problem since the transformer can be built with any turns ratio. The PHY then also has to operate at the right impedance so that it comes out to the ethernet impedance of 50 Ω at the network side. The only way to know what your PHY wants is to look at its datasheet. PHY datasheets should tell you what transformer ratio they require.

POE versus not is mostly a issue of providing center taps on the network side. Transformers can also vary in whether they include a balun (common mode choke) on the network side or not. Note that this also makes the transformer so that it has a network side and a circuit side. Even with 1:1 ratio you shouldn't flip it around since the balun is pointless on the circuit side. The purpose of the balun is to keep high frequencies that might be present on the board from leaking out via the cable and therefore emitting too much RF.

TLA-6T127LF(-T) supports POE in stock.
http://www.digikey.ca/product-detail/en/TLA-6T127LF-T/445-6788-1-ND/2523609

But the extra coil can be terminated to ground. But no problem if open as it is common mode without much different load.