Yes, context can help. Also, this is an early lesson in the course. Later, we learn about the accusative, so that will be able to say, "La viro ne havas laboron." (The man does not have a job).
As in English, Spanish, Portuguese, etc, you could use the present progressive tense (here I believe it would be "la viro ne estas laboranta") to express that he isn't in the process of working at this moment, perhaps because he's procrastinating. But I don't know if there's a specialized word or phrase that would have a more narrow meaning indicating habitual work, ie employment.
My question would be similar, but in a different setting: if I want to pick someone to work on a project, I could say "he is not working" meaning he is available to work on the project, or "he does not work" meaning he is lazy and not someone I would want to work on the project.
Don't worry, your browser is just fine. Some of these sentences simply don't have sound. I think I remember that one of the creators saying that the voice saying the sentences was not a TTS, but someone was actually recorded saying the sentences. Since there is probably well over 1000 sentences in the entire course, I wouldn't be surprised if not every single sentence was recorded.
I believe it's originally from the Latin "laborare," which is where labour comes from. So you are right.
The thing you said translates to: The man is not a work. Esperanto is influenced by Romance languages, Grammar wise, which have no Present Progressive/Continuous. So, you would say: La viro ne laboras, which means both: He doesn't work/ he isn't working. It is better though to say: La viro ne havas laboron, which mean "he doesn't work/ he has no job"
Spanish and Portuguese are Romance languages that use the present progressive quite frequently. (eg, "Yo estoy cocinando la cena." | "Eu estou cozinhando o jantar.") French, another Romance language, also has a phrase ("être en train de faire qqch") that serves this purpose (eg, "Je suis en train de cuisiner le dîner."), although it's not used as frequently as in Spanish, Portuguese, or English.
Yes, both are correct English translations of the Esperanto sentence "La viro ne laboras".