Gem
Since you're using Rails, I would recommend using the Twitter gem by sferik. This gem interfaces directly with the Twitter v1.1 API, and has all the hooks you require:
Followers - client.friends("username")
Favourites - not sure
Retweets - client.mentions_timeline
I'm not sure how to only return time-constrained formats for this gem. Failing a native ability to only pull the required data, you'll have to pull all the data & split for the required time period. However, this will very inefficient & expensive
JS
This is the most efficient & "natural" way to access Twitter data in Rails. Failing that, there is a JS method called Twitterfetcher, which basically parases a Twitter Widget & allows you to display the content as required