The Significance of Phoenix's name: a phoenix is a mythical bird who is born from the fiery ashes of its predecessor. After Beef dies and his body burns Phoenix's career starts to rise.
Why does Swan's Deal with the Devil specify that he has to watch the recordings of himself every day? Because cruel irony is what the Devil's known for. Swan was such a narcissist he first decided he'd rather die to avoid seeing himself grow old, and then made a dark pact with a contract written in blood. He made a Deal with the Devil to avoid having to see his beautiful face and voice grow ugly and aged, but because of that clause he has spent the past 20-odd years doing just that in the most literal way possible.
"Goodbye, Eddie, Goodbye", written by evil record producer/musician Swan, is ultimately a song about how people in the music business can profit from death, though the song tries to dress this up as sweet and sentimental.
As pointed out in the Awesome section, Swan seems so calm and collected when Winslow has a knife at his throat after Swan realizes he's alive after bombing the Juicy Fruits. It's until the twist, when we discover Swan did a Deal with the Devil, that we know why. If Winslow had tried to kill him then and there, he wouldn't have died. He has no fear of death at that moment because he knows he can't die due to being "under contract."
After agreeing to the terms of Winslow, Swan has Phoenix placed as a background singer, saying, "She's perfect. But you know how I abhor perfection in anyone but myself." Earlier, when Swan is adjusting the controls of the synth system to create a voice for Winslow to speak with, he does it while Winslow's singing. However, the singing voice isn't Winslow's from the start of the film, but is Paul Williams' singing voice. Then it makes sense why he says, "Perfect" when he's finally dialed in the synths to create the new voice for Winslow: Because it is literally Swan's voice.
In a movie about music Winslow is a sent to a prison that is called Sing-Sing which is a real prison but it's name has connections to music.
So why does Phoenix seem to have such a hate-on for Winslow? Why would he corrupt Phoenix? Because he liked their music. With other performers, you can see that he doesn't care, that they evoke no emotional response in him. But with Phoenix and Winslow, Swan recognized talent that might rival or even surpass his own "perfect" talent if nourished and given the chance to flourish. Because of his obsession with his own perfection, particularly in light of the fact that he sold his soul to maintain it, well, he couldn't have something like that, could he? The real question becomes just how often has a situation like this happened with Swan getting off scot-free?
Swan mocks the Phantom for not reading the fine print on their contract, but fails to realize something important—Winslow predicated his signing upon his cantata being performed as he wished ("I will write my cantata, but you'd best play what I write.") and Swan merely pricked Winslow's finger and put his own stamp on the contract without bargaining or disagreeing. A verbal contract which existed outside of the physical one, though it counted as part of their terms. "All articles which are excluded shall be deemed included." Without Swan realizing it, Winslow was empowered—literally—to restore it and ensure it was played his way, as per the terms of their agreement. It really did protect him.
Fridge Horror
The teeth removal program was sponsored by the Swan Foundation which means after having Winslow beaten and thrown in jail Swan was still torturing Winslow.
The Juicy Fruits barely escaped the car explosion but what about the women on the car with them.
The pain Winslow must be going through after getting half his face damaged. No wonder why the guy is seen taking so many pills.
