Does Alpha USA already have copyright permission to use the suggested video clips? Or, do I need to work out the copyright issues on my own?

Alpha USA does not hold a license that will cover everyone who gives live talks on the Alpha course and illustrates their points using movie clips.  We rely on the churches, organizations and people running the Alpha course live to obtain their own license or permission. The license we hear is most often used can be found at CCLI: http://www.ccli.com/  

We also found a document by Stanford University that gives an outline on this topic.