Both interpretations of the sentence are correct, but convey subtly different meanings. The meaning could have been clarified if they had added commas and chosen "which" or "that" appropriately.
The first interpretation would be more obviously the intended meaning if "that" had been used instead of "which", and repeated:
He asked a lot of questions that were none of his business and that generally managed to annoy everybody.
This implies the meaning that the many questions he asked that were none of his business also annoyed everyone. In other words, everything after that is extra information about the questions he asked. It does not imply that he was generally annoying in other ways, but that his questions were generally annoying.
The second interpretation would be more obviously the intended meaning if commas and an extra "he" were added:
He asked a lot of questions, which were none of his business, and he generally managed to annoy everybody.
Here, the questions he asked are not the reason why he annoyed everyone. The fact that he asked questions and the fact that he annoyed everyone are separate statements; "which were none of his business" is there to add extra information about the questions. (We can still imply, however, that him asking lots of questions that were none of his business was annoying.) In this interpretation we can swap the order of the sentence, so the following would mean the same thing:
He generally managed to annoy everybody and asked a lot of questions which were none of his business.
I think that (2) is probably the intended meaning given the phrase generally managed, which implies a wider scope for the annoyance than the questions, and the use of "which" instead of "that". However, as written, it is ambiguous as to whether "generally managed to annoy everybody" refers to "him" or "questions", so both interpretations are valid.
Both alternants are plausible, though it’s not about 'compounding' two clauses, but about a coordination of two verb phrases.
 He asked a lot of questions which [were none of his business] and [generally managed to annoy everyone].
 He [asked a lot of questions which were none of his business], and [generally managed to annoy everyone].
In , the coordination is of the two bracketed verb phrases. In this reading, the criticisms were entirely of his questions, which were none of his business and annoyed everyone.
In , the coordination is of two different verb phrases, in brackets, and in this reading, the criticisms are of two separate things: his questions for being none of his business, and him for annoying everyone.
In writing, the two meanings should be distinguishable if a comma is inserted in  before "and", as shown. But in speech there would be no way of distinguishing the two meanings unless there was a pronounced pause before "and" – but even that would not be a foolproof guide and ambiguity may well arise.
Both are correct sentences.
We can interpret "generally managed to annoy everybody" to describe him or to describe the questions.
At the party I met a pretty girl who was Italian and was wearing a pretty dress. The Italian said that she liked my pretty dress.