What they were looking for was actually a nominalizer. You cant say the cause was a verb, you need to nominalize that verb.
The cause for my bad ankle was excessive run = bad
the cause for my bad ankle was excessive running = good
When pairing a nominalizer with だ、である、or です, it has to be こと and not の. This is so it isnt confused with the explanatory の.
Second one is a bit tougher. So we are talking about a something very general rather than a set incident and its not explicitly told that the mother not cooking is the cause for the father having to cook. The first clause is more or less setting the circumstances for the general fact that the dad makes the food.
When describing causes and reasons for things, the ないで and なくて are interchangeable, but in this case, its just saying that in general the father cooks and the mother doesn’t. Its not saying the father has to cook because the mother doesn’t.
Honestly its pretty confusing and I think I may have gotten it wrong too, but I think the あまり and よく are the best signs since it sorta gives a more general feeling to it. Honestly I’m not even fully sure that I get the whole story but I’m confident enough to post.
Sources:
https://www.alc.co.jp/jpn/article/faq/03/99.html
https://www.alc.co.jp/jpn/article/faq/03/2.html
if youre lazy, heres the line im looking at for how to use ないで
あるいはある動作が行われる背景となる付帯状況を表します