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.
Yes there is, but it doesn't tend to be used as much in Esperanto as it is in English. If you want to emphasise the continuous nature of an action, you can say things like: "La viro estas laboranta" ("The man is working"), "Mi estas pensanta" ("I am thinking"), "La verkisto estas verkanta sian dekan libron" ("The author is writing his tenth book"). But usually, Esperanto sticks to the simple present tense.
Yes, both are correct English translations of the Esperanto sentence "La viro ne laboras".