Quotes from real people and books are generally considered fair use so they can be used without paying anyone. Song lyrics and poems are a gray area. Some people will tell you you can use portions of song lyrics - a line or two - but others will say you can't use any without permission.
If you're not using the whole song, you'd probably be able to get permission to use it without paying anything. It'd be free publicity to the owner of the lyrics - most likely the record company. If you're using the entire song, that's completely different and you might be require to pay for permission to use them. How much and whether it's a flat fee or royalties will depend on how it's negotiated.
Unless you're a NYT Bestselling author and the publishing house really really wants you, the author will have to pay this out. Publishing houses aren't going to cut into their own profits unless they really have to.