Strictly speaking, "public transport(ation:US)" is trasporti pubblici, while mezzi (= "means") refers to the vehicles. But even in English we sometimes confuse the two, as in "the buses are disrupted today". European city transport is typically multi-modal (buses, trams, metro, etc.) so it is natural to refer to them together with words like mezzi.
"Mezzo" is the equivalent of the word "means" in English. A "mean" can be a middle, or average (as in the difference between statistical "means" and "extremes"), or it can be a way or mode of doing something (as in a "ways and means committee"). The Proto Indo-European root of both words, "*me-," means "between" - which makes sense in all cases.
A similar English word would be "medium," which can also signify both "middle" and "means," and comes from the same root.
Excellent question. Answer 'Yes!' In fact it would be less likely to cause confusion. Il trasporto pubblico = public transport as a generic concept. I trasporti pubblici refers to the whole public transport system of a geographical area (anything from a town to the whole country). It's what you'd ask about at an information desk.
Mezzo pubblico is short for mezzo di transporto pubblico (means of public transport) and in the singular it can refer to a public service vehicle. Duo's sentence actually implies "we want buses, trams, etc." and is as much a complaint from citizens as a sensible query from tourists. Better to ask dov'è l'autobus per ...?
The plural is used for generic transport (-ation in the US). Hence i mezzi di trasporto su ferro = rail transport.
"Trasporti" is more formal, so in normal conversation you're more likely to use "mezzi". According to google "trasporti" is losing popularity even in writing: https://trends.google.com/trends/explore?date=all&geo=IT&q=%22mezzi%20pubblici%22,%22trasporti%20pubblici%22
Well, that's complicated, because the italian verb "volere" is used with different meanings, and this leads to different english translations (and apparently Duolingo cannot deal with this).
you can use "volere" if you want something (or something to happen): "io voglio una mela" (I want an apple, a little bit unpolite, but correct)
you can use "volere" if something is needed: "ci vuole un cacciavite" (it needs a screwdriver).
you can use "volere" if a specific amount of time is required: "ci vuole mezz'ora per andare al lavoro" (it takes half an hour to get to work)
As you can see, completely different meanings (different translations), same italian verb. This is pretty confusing, I suppose (I'm italian).
For this specific sentence, the correct english verb is "to want" :-)
Hope this helps.
In the US, "public transit" and "public transportation" are in common usage. (If you google the terms, you can see that both are also used officially. "Public transportation" is the more common term.) "Public transport" is apparently a British usage, and is not generally used in parts of the US where I have lived. [US Native English speaker]