> そのことで日本のだれに何をいわれようが、女房が宇宙任務を **果たしたこと** には変わりがなかったのだ。

I'm not sure what role こと plays in this sentence. I heard that it sometimes
means like a rule, or something that must be done.

My nearest guess here as to a translation here is:

> No matter what anyone in Japan would have said, the fact that my wife
> managed to become and astronaut would not change.
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grammar
Use of こと in the following sentence with 果たしたこと
> 宇宙任務を果たしたこと

宇宙任務 sounds a bit weird and ambiguous, but this phrase means "the fact that
(she) have completed her space/astronaut's mission."

I think the other part of your translation is OK.
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> 宇宙任務を果たしたこと

In this sentence こと is turning the clause into a nominal, making it so it can
be used like a noun. So it becomes "the fact that she completed her space
mission" (as was answered earlier).

There is another grammar form where こと can be used to make a very formal
command or declaration of a rule that is almost always written. Sometimes it's
written on signs like in

> ドアを閉めること。
>
> Shut the door (after using it).
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answer