The third rail has actually been coping admirably this morning - I can’t recall reliability quite like it in these weather conditions. One of the only notable third rail issues currently ongoing (at Cookspond Viaduct, near Dormans station) is probably not weather-related, and the built-in backup of third rail power feeding equipment means the trains can mostly still run.
In terms of weather, it’s been the case that level crossings, points and indeed train doors have all been the worst affected by the ice and snow. This has occurred both north and south of the Thames, pretty much in equal measure, including on other networks as well.
However, Southeastern, Southern, GX and Thameslink have also had to contend with a number of broken down trains, some of which have had failures which probably didn’t get helped by the low temperatures, but also may well not have been explicitly caused by them. The irony is that today’s most disruptive breakdowns have included a number of engineering and railhead treatment trains, which have probably had a combined worse impact on the service than if they hadn’t run at all. I’ve mentioned the train at Herne Hill, which has caused absolute chaos, but special mention also goes to the MPV near Wye which decided to have a disintegrating brake valve at a random level crossing, the rescue train not being properly prepared, and the combined train then issuing a strong burning smell.
Oh, Thameslink has more than just “a set of crossovers”. Way more. But when you have trains stuck across junctions in south London, with only a comparatively small number of drivers currently being able to divert away, it quickly becomes bonkers!