Is it appropriate to email a professor during summer for a difficult question?
Let me try to answer some of the questions in the body of your post:


  In general, how do professors react to this? If you are a professor (who is not someone who doesn't care about students) please comment below. Are they annoyed, or are they ok with it/delighted? What if they are busy?


Of course the answers are "it depends". What does it depend on? Here are a few factors:


The tone of your email: of course, people tend to respond better to polite requests than something that seems demanding or pushy. You say the professor "might well be busy". I can guarantee you that he is busy. He may or may not make the time to answer your question nevertheless, but you should make clear that you understand he is not under any obligation to do so.
The difficulty of the question: you say your question is "rather difficult", but what is rather difficult for you may not be for a professor who is expert in the subject. If your question really is something complicated that will take a long time to type an answer to, people will be less inclined to spend that time just to satisfy your curiosity.
The clarity of the question: if you choose to write and ask, you should be absolutely sure that your question makes sense and is clearly and unambiguously stated. Nothing is more vexing than putting in time to answer someone's question only to have them change what they are asking.
Personality: some people just enjoy answering questions more than others; some people guard their time more closely than others. 


Speaking just for myself, if I received such a question over the summer that was clearly written, not overly complicated, and phrased in a polite way, I would be pleased (maybe not "delighted"), and if I had time, I would try to answer.