Nice question. Reading up on it, "linguaggio" seems to refer to the whole system of human communications, or rather the system of shared conventions permitting humans to communicate: thus it extends beyond the concept of "lingua", which is the subset related to speech and writing, to include gestures, expressions, glances, and so on. Thinking about the internet, emoticons are what brought written communications to the level of "linguaggio". Interestingly to explain all that the Crusca website quotes a Francophone Swiss linguist, Ferdinand de Saussure.
As for the first point, you could use "col" too as far as I know, it's just rarer to join con with the article compared to other prepositions for some reason.
Here is a forum discussion on the difference between lingua and linguaggio which might help.
It may be common, but the words are different, so different words are likely used in Italian, too. I don't know why some people have a problem with translating clear sentences the way they're written, instead of insisting that it CAN be something different. Some Australian wanted their evening "tea" to be accepted as "dinner." Well, we're having "brunch" here, let's say, but we don't expect it to be accepted as either "breakfast" or "lunch" on a site catering to people from various countries. Duo barely manages standard English.
nerevarine: Just physical cababilities? How about ... well...language? "We're able to speak Italian." Especially in it's opposite: "We're unable to speak Italian". How about situations involving permission? "Are we able to speak Italian in our classroom?" Ok, Can and Able aside (well we know how THAT ended don't we!), I think removing 'the' from the sentence is more in keeping with the meaning of 'linguaggio' since my understanding of it vs. 'lingua' is that it refers to a specific jargon or language of the street, everyday language. So there's the Italian language -lingua- and there's say the language of Calabria, Sicily, Naples, etc. -linguaggio. The complete phrase used is: "nel linguaggio quotidiano". At least that's how it was explained to me by a native.
I agree that removing or adding the article before "language' changes the sentence. Without an article, it is simply referring to language in general.
This may be entirely subjective, but "We are able to," in this sense, just hits the ear wrong. It sounds like someone who learned English from a book. For example, "We're unable to speak Italian," is technically correct, but it would much more commonly be rendered as, "We cannot speak Italian.'
nerevarine: Good point below. In thinking about it, another point comes to mind, and that is that able/unable strikes me as more emphatic than 'can/cannot'. "We can't go to Italian today" sounds like just a general statement vs. "We're unable to go to Italian today" which to my ear at least sounds more emphatic, implying a more serious situation.