Basically, you're talking about muscles, just like any other muscles. And just like any other muscles, regular, structured, exercise will make them work better.
But since your vocal cords are pretty much a finite length, you're probably not going to gain more than a couple of notes at each end of your range.
Vis a vis singing along with music playing through headphones, make sure you keep one cup off your ear (unless you're singing through a microphone) or you'll probably learn to be someone who has good voice control but who sings off key.
When you sing, you are hearing two separate components: bone conduction - where the sound is transmitted through the bones in your face to your ears, and air conduction - where the sound comes through the air to your ears, which is how you hear most other sounds (and how others hear your voice).
If you're listening through headphones over both ears (and not singing through a microphone), all you are hearing is bone conduction, which does not give you an accurate depiction of your voice.
This is why you will sometimes see a singer "cup" their hand behind their ear while they sing, as it improves the air conduction of their voice...