If lots of people have tweeted tweets containing exactly the same phrase (e.g. a marketing line or URL), is there any way to randomly pick one user who did this? If so, how? Please help
You might want to check out http://twiveaway.com They allow you to select a random 'winner' out of the people who have retweeted your tweet.
Oops! I didn't notice that. You can also take a look at http://www.tweetswin.com/new/ or http://tweetaways.com/ or http://www.itweet2win.com/ (I haven't tried those myself, but it looks like they should do the trick)
I wouldn't do this, you'll get more out of twitter if you interact with your users. there are many sites that you can schedule tweets, this is most likely what you want, but I would advise against it.
I am not exactly sure how this works. How do you get the message to people who are not following you.
(most - or all - of) those services require people to follow you and then tweet a message in order to enter the contest
the only way to message people without having them follow you or you follow them is to use a too like TweetUnlimited (com) - where you automated the @replies without actually doing a real "reply" and do send messages as mentions. Works great as it does not use the Twitter API as most scripts do, which gives you heavy limits!