... I don't see how Emma did anything horrible here. On the contrary, if she had not asked about the Dixons, it would have been considered rude. People were expected to show interest in such things. If you met an old acquaintance whose friend was recently married, you were expected to make smalltalk about it. Moreover, I don't think Emma asked about it in a vulgar way. The text goes like this:

"She [Jane Fairfax] seemed bent on giving no real insight into Mr Dixon's character, or her own value of his company, or opinion of the suitability of the match."

Here we are not told that Emma was impertinent. The fact that Jane didn't give a clear insight does not mean that Emma asked for the insight in an impertinent way. I assume she just made the usual (society-wise) questions and took non-committing answers. She wouldn't throw Jane in a chair, put a torchlight in front of her eyes and go "so, Miss Fairfax, what is Mr Dixon's character, how do you value his company, and what is your opinion of the suitability of the match?"

As for Frank Churchill, of course she was curious and she asked questions about him, but I don't think that's a bad thing. Frank was a friend's son, a particular friend's stepson, and if asking questions about a third person in a social gathering is a crime, we are all doomed.