With words like "today", "tonight" etc we usually use "present continous tense" so "Dad, why aren't you going to work today?" would be correct
That would be "go work". with "work" as a very. This is "go to work" as a noun, in the same sense that one might go to school.
You spelled "verb" as "very." Also, I think you are trying to say that "trabajar" in this sentence translated to the English gerund "working." The end of your sentence, "... in the same sense that one might go to school." makes no sense to me. Are you saying that "schooling" means "going to school?" It doesn't in English. In English, "schooling" means the amount of education that you have had.
I wrote, "Dad, why you're not going to work today?" and it was marked wrong.
I think you mean the verb "accept." In the context of your answer, "accept" means "allow/take in," and "except" means "single out/deny permission/not allow."
Because I have trouble deciding when to translate Spanish present tense into English progressive progressive tense, I imagine native Spanish speakers have the same trouble grasping when to use English progressive tense instead of simple tense. The issue is that the English verb "to do" is a modal verb (modal verbs convey how the speaker feels) and the English verb "to be" is not.
Denotatively (i.e., literally), your alternate translation is correct, but the connotative English meaning of "Daddy, why don't you go to work today?" is modal and thus the child is suggesting that Daddy SHOULD go to work today. Conversely, the connotative English meaning of "Daddy, why aren't you going to work today" is "WHY isn't Daddy going to his job?" Thus, even though the Spanish verb "va" is not the one of Spanish modal verbs ("poder" and "deber"), "va" still lends itself to being translated as English progressive tense rather than as English simple present because the concept of "go" in itself is progressive.
If some native Spanish speakers can confirm and supply the reason why, from a Spanish mindset, this present progressive English sentence should be translated into simple present tense Spanish in order to keep the same meaning, it would be much appreciated.
cordwainer9 supplied confirmation above.