You can easily add "la" to "casa" but it changes the meaning from "home" to "the house".
As far as why "trabajo" needs "del" rather than "de"--I'm not sure. When I looked up the phrase "de trabajo", I found that the phrase is often used as a modifier for other nouns like an adjective. Perhaps, that's why???
Bruce, it is because trabajo meaning "the job" or "work" is a masc. noun, so must use the contraction for de+el = del.
You were correct about la casa, but of course the fem. article cannot use the contraction, & the expression "at home" doesn't use the article. I guess Duo is teaching us "leave (from) home" doesn't need the preposition either. I see Duo feels the "from" is "understood," because it wasn't wanted in the lesson sentence (although it should not be counted wrong, IMO).
As an English speaker, I would not say, leave the work. The definite article is not used. Spanish is like this as well. In general, the, a, an are not used, apart from in specific situations. For e.g. when you want to emphasise the exact place. This isn’t necessarily going to help you sort this one, but I have gone through various similar statements & can’t find any rules to them. Some have the and some don’t. I really does feel intuitive.
In english asking someone( Are you leaving for work at 5 ) is not the right english because the word LEAVING means to "go away" so instead we say Are you going to work today at 5 (te vas a trabajar a las Cinco)"vas" is nothing but you go. This sentence can also means i am going to do some work today at 5 ? But if you're specifically asking someone are you going to your workspace or office at five in spanish it would be (vas a la oficina a las cinco) Conclusion leaving and going have two different meanings to it in both spanish and english. Kindly look into it
Your question got me thinking of "implicit bias." Thanks for the wake-up call. While personally I have done shift work my mindset is that jobs are, as Dolly Parton sang, "Workin' 9 to 5" as in you leave (Salir de) your house to get to work at 9 am and then you leave (Salir de) your job/work at 5 pm and go home for the evening. But construction, factory, and service jobs (janitorial, food service, health care, bus driver) often do start early, so that you would leave FOR work at 5 am. On a different note, there are a number of "transitive verbs" including salir de, but also irse de, marcharse de, and partir de. You can read more here: https://www.spanishdict.com/translate/to%20leave
Just don't overcomplicate this one.
"A las cinco (horas)" A La/ A Las/ A los---don't change.
Only "A el" changes to "Al" as in "Al trabajo"