It is grammatically correct, but not really a correct translation of the sentence. To say "but no shoes" leaves out the "ho." So it is correct, but not as a translation to this sentence.
The "partitive article," in this case "delle," indicates an unknown amount of something. Here it means "some socks." It is often used in Italian where it would be omitted in English. "I have socks" really means "I have some socks" but not a specific number. There's a fuller explanation here: http://italian.about.com/od/grammar/a/italian-partitive-articles.htm
I agree. I hope you submitted the feedback "My answer should be accepted." They take it seriously, and changes are often made. The program is in BETA because they want this feedback.
It would be correct Italian. The sentence with and without article doesn't change much, without article it's more natural. I think that's the same for English, am I right?