I think most people would just say "arrive home," but it's not accepted. Reporting.
Your answer is the best translation, now edited, thanks for the comment.
Sure, my pleasure.
You'll do the dishes after you arrived home is accepted, but with a Typo? I don't see the Typo.
"arrived" has an extra "d" compared to "arrive".
I don't think you can use the simple past here for an event in the future, so it has to be either simple present or present perfect ("after you have arrived").