I host a web site using LP's basic hosting plan, and the client is now asking me if I can provide detailed reports on what files are downloaded how often, and if I can get them auto-e-mailed to them on a weekly basis.

I've looked through cPanel and although Webalizer and the other tools seem OK, they're not as detailed as I would like, nor are they as automated.

Can anyone recommend any tools/programs I can install on my site to monitor downloads in detail and provide me with customized reports according to my spec?

Please make sure in future posting to keep one question per thread, as it makes it difficult for fellow forum members to track if you have been replied to or not. 

Have you looked at Google Analytics?  I haven't played much with auto e-mailing through it, however did just run into a little hack you can do to get file download numbers for it.

http://blog.lunarpages.com/2009/04/28/how-to-track-file-downloads-in-google-analytics/