Very well, both cases. In the case of finding out how often a certain user mentions others you could use the user_timeline resource without needing authentication. You could fetch 200 tweets by a user a time and then make use of the since_id parameter to keep fetching older tweets. For every tweet you can then check if it contains a mentions.
As for the other case, I'm not sure. You could use the search API with a search string of "to:username" for @replies or a search string of "username" which would yield tweets by that user, as well as @replies and @mentions. Twitter search only gives results for the last couple of days though so I doubt that this would help you.
Good luck!