python-twitter should cover the first four requirements. I’ve used it before, and it’s fairly easy to start developing with it. For leveraging Twitter’s streaming API, I would recommend tweetstream. It’s a fantastic Python module that grabs tweets in real-time as they are posted. Based on whether you have gardenhose/firehose access to the twitter stream, you’ll only get a small fraction of tweets posted. With tweetstream, you can also provide a list of search predicates to filter specific tweets that you are looking for. I used it for a project that involved mining tweets over an 8 hour period and it worked flawlessly. Both of these modules should be available through Python’s easy-install.
EDIT: I don’t know what you intend on doing with Python/Twitter but if you do plan on capturing a lot of tweets, keep in mind that Twitter receives myriad tweets in languages besides English. Remember to encode everything properly.