Translation:She is sick. She should not go to work.
In case it isn't obvious due to the inadequate tooltips, "生病" is a verb meaning "to become ill": https://en.wiktionary.org/wiki/%E7%94%9F%E7%97%85#Chinese.
Are there other examples of 生 being used to mean becoming/acquiring a state, or is this construction not productive?
I don't think so, I'm pretty sure 生 has to do with health here, while it does mean "to give birth/ life" so you could think of this as "give birth to illness" it's not really used that way in other compound words - it usually has to do with life and health. It's 了 that indicates the change in situation; from not ill before to ill now.
It doesn't feel good to have to google the meaning of 生病了, as Duolingo so helpfully provides me "give birth to sick".
I suppose 不应该 could also be translated by "doesn't have to" in this context.
There needs to be some punctuation or connector such as "and" or "so" or because
"She is not well and shouldn't go to work" corrected to "She is unwell and shouldn't go to work."
Thank you :) I am not a native, and exactly this kind of details helps me very much improve in English. :)
I don't understand the meaning of 了here. Does that mean "becomes" ?
See the grammar explanation for this lesson. That is "changing situation". Means that she wasn't ill but situation is changed and now she is ill.