This is NOT "accidental damage", and is in fact something that isnt supposed to ever happen, within or outside of warranty, and I would be surprised if they didn't repair it for you.
This is accidental damage, these power bricks can and
will get warm. Any and all power adapters get warm. Power adapters are engineered to dissipate heat, and the magsafe specifically will turn off if they get too hot ( which could explain why it yours was off and cold in the morning). They were designed to be used in a well ventilated area.
Your extreme negligence regarding your poor treatment of the adapter and macbook is your fault. A hot adapter placed on top of the plastic macbook is clear recipe for disaster. Something that can get hot should never rest on top of a plastic macbook.
That is an Apple problem. No magsafe should be that hot to do that much damage. they owe you a new top case and palm rest and screen frame at the very least, a new MacBook at the best.
Why is Apple responsible for misuse of products?
"The power adapter may become very warm during normal use.
Always put the power adapter directly into a power outlet or place it on the floor in a well-ventilated location. If you are using your MagSafe adapter in a poorly ventilated area, or if the MagSafe adapter is covered by a blanket or other form of insulation, it may turn itself off in order to prevent damage to the adapter." -Apple kbase article TS1713
Thank you for your comments. I will bring it in to the Apple Store tomorrow to determine if it will be covered. Any advice for what exactly I should tell them to have the best chance of getting this covered under warranty? I am going to be honest with what happened, although I know that it may not qualify for the 3 year warranty. Regardless, I hope that Apple stops making power bricks that get hot enough to cause this type of damage. I'm just glad it did not start a fire or destroy anything else.
Be honest, tell them exactly what happened and hope they will go out of their way to help you. In my opinion, It's still your fault.