That mostly reflects the English usage. I would say, "I'm going on the Internet tonight", or "The internet isn't working for me" but "There appears to be no internet this evening". No, it doesn't seem to be that regular.
The reason that it does not have a definite article in the Italian is that in Italian it is a Proper Noun like a name. And such do not always require the definite article. http://italian.about.com/od/grammar/a/article-with-italian-proper-nouns.htm
I agree with you completely, and, obviously, would not say "We have no the internet tonight" :-). However, if we want to be really technical when we speak about the Internet, we should say "We do not have connection to the Internet tonight" or "We cannot access the Internet tonight". Thus, I thought at the time I've written the answer and the comment that "We do not have the Internet this evening" should have been accepted.
It is not inconsistent with the word, rather it is just the nature of articles in English (and other languages as well). We do this all the time, for example: "The apples are fresh." "We have no fresh apples." "I want the fresh apples," which has a slightly different meaning to "I want fresh apples."
It appears from these examples that Italians use "Internet" to mean "Internet access" or "Internet connectivity". I have never heard any English speaker use the word "Internet" like that. If we use "Internet", it is always with "the", i.e. "the Internet". I think "Internet connectivity/access" is what you're looking for here.
No, I think it is just an error. There is no 'accesso' or 'connettività' in the sentence. Literally translated, the Italian version is 'Not we have Internet tonight/this evening.' Constructed properly in English: We do not have the internet tonight - or - we have no internet tonight. [The article is in the Italian sentence, just not immediately before 'internet.']