LAST EDITED ON 08-18-11 AT 09:12 PM (EST)Let's answer these in a different order than asked.
Songs are published by publishing companies that in turn belong to one of three performing rights organizations in the U.S.: ASCAP, BMI or SESAC. (Canada used to have two, but they merged into one, called SOCAN, in 1989.) The performing rights organizations license public performances of their members' music to network television, cable channels, nightclubs, stores, restaurants, politicians and other places where the songs will be performed with a public audience (except for dramatic performances, such as in a play, which follow a different regime).
"Curious, how much of a tune has to be performed in order to owe a royalty? A full 8 bars? Or would a one or two bar snatch qualify? And would it have to be the words with melody, or would the words of a song without melody (or horribly sung) qualify, or what?"
I believe virtually any amount, including a one or two bar snatch, would require payment of a royalty under the PRO contracts. And it counts even if you just hum the song and never sing the words, since someone composed and published the music -- unless it's a tune in the public domain, such as an old folk song.
"And does Weird Al Yankovic (for example) pay royalties for his parodies?"
The authors of the songs that Weird Al parodies do, in fact, collect royalties. Similarly, the authors of songs sampled in other songs collect royalties for the use of the samples. There have been some acrimonious disputes about the value of a sample. In one of the most egregious cases, a music publisher threatened a rock group (that used sampled music in the background and had already agreed to give the authors of the sampled song full composing credit for the music) with refusing permission for use of the sampled music unless the authors of the original song were given 100% of the publishing credit. Because the song was becoming a hit, the group felt like it had little choice but to agree -- because the song would have to be withdrawn from release if they didn't. And that's why the sole composing credit on the song "Bittersweet Symphony" by the Verve, which was written by Richard Ashcroft but used a sample from "The Last Time", reads "Mick Jagger-Keith Richards".
"There are a lot of songs that might have the same lead-in or similar progressions (I.e. almost every dang country western song before 1950), so there have to be limits."
If the song can't be identified, how would a publisher be able to prove that it owns the song? But that's not going to happen very often on BB13.
"And, for example If a Herb Alpert melody was played on a piano instead of a trumpet, would a royalty liability ensue?"
Sure. It's a public performance of a published song.
"Finally, how much of a bill has Dangerous Sagebrush Dan racked up for his birthday dittys?"
I haven't seen any public performances of them, so I'd say ... about $0.