On the music side, I don't believe an artist needs to receive permission to perform another artists songs live. However, if someone records the performance and distributes it, then you get into royalty issues.
Do some research over at Harry Fox for more info. They are the main clearing house for this type of thing:
http://www.harryfox.com/index.jsp
Showing movies out on your patio is clear violation of copyright. Its a public performance, and your making money off the partrons. You could probably get away with it for quite a while, and a cease and desist letter is the likely response if you are caught. The city might take issue with you opening a 'theatre'.