1. "I look forward to his response." Both sentences are correctly written, but the simple version is always best. Only add the explanation at the end if the reader might need it to understand your meaning.
2. Both sentences are correctly written and sound polished. There is a slight difference in meaning. The word "flair" in option b connotes that the person's writing has a sort of flashy, attention grabbing style. Unless you are talking about someone who will be writing advertising copy, or entering a contest for writing the best short story, then option a is probably the better choice for your purposes.
3. "I didn't manage to follow well that part of the conversation." This sentence means that you found the conversation difficult to understand. I think this is what you meant to say.
"I didn't manage to capture well that part of the conversation." This sentence means that you are trying to portray or describe the conversation/movie/lecture to someone who was not there, and are finding that part difficult to describe in a way that makes the other person really imagine what it was like. We might say that a really good photograph really "captures the moment" if it makes people understand what it was like to be there at the event. We could also say that of someone's description of the event, but I don't think that is what you meant.