Is it possible to count how many times some URL was tweeted in some period of time?
You can use the Twitter Search API and search for URL's, then examine the Tweet objects to find the ones in your date range. Use the GET search/tweets call. This is limited to the past 6-7 days in their index.
You can also write an app to watch the Twitter stream for the URL over a period of time, then count the occurrences. This can be done using the Twitter Streaming API using the POST statuses/filter call. Specify the string you are watching for and matching results will stream.
In Ruby, for example, you can do this using the Twitter Gem. It's best to search for known URL-shortened versions as well as the full URL. You can control this somewhat by registering the URL with several known URL shorteners in advance, so you know what to watch for.