No, it actually only means 'on the way', not related to road work. In general דרך is not a direct translation for 'road', it's more like way/path and applies to figurative speech like 'way of the warrior' (Bushido) more commonly than actual roads (although in some contexts in does mean literally a road). The common word for actual roads is כביש (kvish).
Just throwing this out there in addition to what you wrote because it helps me remember ״דרך״ and emphasizes the figurative meaning- in Orthodox Judaism in English speaking countries the (quite common) phrase for someone who has left Orthodoxy is OTD or "off the dereck" (so "off the path"). It was the first way I ever learned the word and maybe some others have heard it too. I think in terms of figurative meanings it can really help to have examples since just saying road or path can be confusing or unclear. If nothing else, it helps me.
Note: There has been a lot of confusion about this sentence here, so I'll try to explain it. A disclaimer: this is a strange sentence, I can imagine very few real contexts where it can be used.
First of all, it does not mean any kind of road work. The context I can imagine is: Do you work during your commute? Actually that's a good translation that should IMO be accepted (I'll try it next time). The other context could be that you're telling someone that you're going to be traveling by public transportation, and they would ask if you plan to work during the ride—but that would likely be in some form of future tense.
In all of the above contexts, when saying such a sentence, one would emphasize the word 'oved' (עובד), not the road.
So could it also be used for someone who drives a lot for work? Like the example I thought of was say a home care nurse who works a lot "on the road" versus most nurses who work in a hospital or clinic?
Or would this be more literal as in when I was a kid I often didn't get all my homework done at night so I'd literally be doing it in the car on the way to school. I guess what I'm trying to say is it more figurative as in someone who drives or travels around a lot for work or more literal in that you actually literally work while driving or commuting? Or would maybe both work? I guess intially I was thinking in very US-centric terms where long commutes are common? But then commuting is also somewhat different than the nurse example so ahh, I've officially just confused myself more. All I know is we seem to have established this does NOT mean something as literal as someone who is a road worker (like in construction or someone who say, pours the actual gravel or blacktop on the road).
It cannot be used for working in the field or remotely. I guess someone who doesn't work from a fixed location (a field worker) would be עובד שטח, or 'he works in the field': הוא עובד בשטח. A less common way (can only be used in certain cases) would be עובד בדרכים.
The second usage you asked about is correct though, it's precisely for stuff like doing your work on the way to [school/work].
I had to choose from a set of words where it could only accept "are you working on the road" but it does not feel like a natural translation. Yes it could mean are you working in transit but that isn't a common way to understand the English sentence that I had to choose.
It isn't the audio, it's unfamiliar words in a non native language. They studied it, that language students always think it sounds faster. I agree it should be slowed down. The actual words you can hear individually using Memrise Duolingo Hebrew vocab course. I think the sound is much clearer. You can also get an app like Reverso, it's a really good computer voice, you can slow that speed in settings & it shows transliteration (pronunciation) so you can get a feel for how it should sound. I like Linguistix YouTube channel, he's a pronunciation coach, like a linguistic specialty field & he's got Hebrew, Brazilian Portuguese and English pronunciation videos.