"He walked back home crying."
Translation:Hij liep huilend terug naar huis.
Yep archaic (and grienen is nowadays more or less insulting (e.g. Wat loop je nu weer te grienen).
Question about word order. Can you say, "Hij liep terug naar huis, huilende"? I tried it and it was rejected. (Also, it's frustrating and often confusing when you get something wrong, but they give you a suggested correct answer like "Hij liep wenend naar huis terug."
I realize the mods probably can't do anything about the apparently random corrections Duo gives you, but it makes it hard to figure out a pattern.
Yes, the random corrections are very annoying. The developers are working on it.
As for your question, that might have been fine in the 19th century or so. However, it would sound extremely quaint and archaic in modern Dutch.