What are the connotations of "ignorant?"
Well I am taking the bonus part here.
The tricky thing about the English ignorant is that there are two words in German that are written the same ignorant and Ignorant, adjective and noun. On the surface the definition is about the same as in English - with the second one refering to a person of such characteristics.
Both have the dictionary addition regarding usage: bildungssprachlich abwertend. The first word has no direct translation. It's along the lines of used in language that requires a certain (high) level of education. The second however means more or less pejorative.
So there are two German words that sound and look the same that definitely have a negative connotation. So speaking German one would obviously pick a synonym without these connotations if you don't want to be that negative.
Since the plain dictionary definition of the pure sense of the word is almost the same in both languages in stands to reason that those negative connotations carry from the native language German over to English for some speakers.
Finally it would be interesting to see what it was translated from. If it actually was translated from the German term ignorant, you should certainly look for a word that definitely has negative connotations.
Saying "Alice is ignorant of the fine details of etiquette" is not too derogatory by itself; the specification of a detailed context focuses the connotation on simply being unaware/uninformed (note the latter two are even safer).
Saying "Alice is ignorant" is derogatory. Without any constraint you are implying that Alice is unaware/uninformed of things that are common knowledge, and thus she should be aware of.
Of course there are gradations "Alice is ignorant of basic table manners", basically means that Alice is a boor. It's less insulting than the global "Alice is ignorant", but still implies a negative judgment.
Note that "Bob is not stupid, he's just ignorant (about X)" is an idiomatic expression, which has charitable connotations.
In net, the negative judgment is not conveyed by the use of the word ignorant itself, but rather from an implied, or stated, social expectation that the person should know the information that he/she is ignorant of.
The meaning that I get from someone being called 'ignorant' is that they're not only unaware, but they also refuse to want to try to become aware. In this sense, ignorant is also connotative of stubbornness and being too stuck in one's own backward biases.*
In the old Webster's dictionary of 1913 this use of ignorant is shown to have Biblical origins, and is defined as:
- (Theol.) A willful neglect or refusal to acquire knowledge which one may acquire and it is his duty to have.
Webster's 1913 dictionary
The Bible quotes on ignorance:
"They are darkened in their understanding, alienated from the life of God because of the ignorance that is in them, due to their hardness of heart."
"My people are destroyed for lack of knowledge; because you have rejected knowledge..."
Perhaps this is the reason that a common insult to call a racist/bigot is 'ignorant', because he would not only be lacking in knowledge, but he would be willfully lacking in knowledge, and this is an important distinction.
Google Ngram results for ignorant:
*Of course, this distinction doesn't apply to uses of the phrase 'ignorant of', which @Dave expounds on in his answer.
'Willfull/willfully' is a collocation of 'ignorant', found in the phrases 'willfull ignorance' or 'willfully ignorant': https://en.wiktionary.org/wiki/willful_ignorance
In fact, this connotation is so obvious to me now that I've remembered the word 'ignore':
Oxford English dictionaries
Refuse to take notice of or acknowledge; disregard intentionally
The word ignore itself was created in the late 15th century from ignorant, which predates it by about a century. In fact, its original sense as a verb was merely to signify to be ignorant of, while the sense that is currently in use hadn't become popular until the 19th century!
etymonline etymology
ignore(v) - 1610s, "not to know, to be ignorant of," from French ignorer "be unaware of" (14c.), or directly from Latin ignorare "not to know, be unacquainted; take no notice of, disregard" (see ignorant). The original sense in English is obsolete. Sense of "pass over without notice, pay no attention to" in English first recorded 1801 (Barnhart says "probably a dictionary word"), and OED indicates it was uncommon before c. 1850.