Learn English – How to use “while” in the middle of the sentence

I strongly suspect that that is was coined as a literal translation of the Latin expression id est (i.e.) with the same meaning, because the construction makes more sense in Latin.

Originally, that is (to say) was used at the beginning of a sentence, where that referred to the previous sentence, or between the two sentences, just like i.e.:

The first Roman Emperor was Gaius Julius Caesar. That is, he was the first man who ruled the Empire de facto single-handedly, though he was no emperor in name.

In later use, its position has tended to shift around a bit, but it still refers back to the previous sentence or an idea expressed therein.

Your analysis of it is perfectly fine. You could say it is the subject and the that clause an appositional complement to the subject.

This did not happen until 3 am, (namely) the fact that they returned home.

It is of course a fixed idiom, so that the internal structure doesn't matter a great deal any more. Note that it is used in many languages, like French c'etait là que..., Dutch het was daar dat... = "it was there that...".

In these dispositions then has a different reference and significance.

Set off from the sentence with commas, as in your first sentence, then is still a sentential adjunct but it has the meaning “thus, consequently”, drawing a conclusion from the matter which has gone before. Your first sentence may be paraphrased:

So the emasculated society of Europe serves as a warning to conservatives ...

Then preceding an attributive adjective, as in your second sentence, modifies the adjective with the sense “at that time”. Your second sentence may be paraphrased:

The society of Europe, [which was] emasculated at that time, serves as a warning to conservatives ...

In your second sentence the comma after Europe should be deleted.

The then emasculated society of Europe serves as a warning ...