Why isn't there an inversion: Den ting fortalte han os i gar. - Isn't it necessary?
My confusion was, I think, because I was approaching the Danish as a complete sentence instead of a phrase. (Capital + period certainly looks like a sentence, right?) Would "Den ting at han fortalte os i går." be acceptable too, b/c that would be clearer for those coming behind us. I saw no inversion, but couldn't infer quickly enough, I guess, that it meant it wasn't a sentence.
exactly my point. It should be accepted, as the other way around in English is very uncommon, though in Danish the word order is of course fine, I'm sure.
It's an entirely different sentence. Yours is reminding people that he said it.
The correct one is referring to the thing, assuming everyone already knows. The whole sentence is a single object. the "(that) he told us yesterday" is just a description of "That thing".
How would one know that it was "that" thing and not "the" thing from this sentence?
"The thing" would be "en ting", while "den" is "that" (or maybe also "this"), I guess
Yes, you are right. That was by mistake, the rule is still the same, though