Upgrade from iPhone 4s to iPhone 5s? - 1

I've had my iPhone 4s since it was released. Since then, I have damaged it severely to the point where it won't even come on. I paid for the warranty, but I never used it. My question: being that I purchased it when it was first released can I go ahead and get my upgrade to the iPhone 5s? Or would I need to pay full price for it? Would they still take it even if it's damaged? I've never upgraded a phone before. Also, if I did this, could I be able to do so when/if the 6 is released?

At this point, get the iPhone 5S. It's the most recent phone, so it will support everything the iPhone 6 will have.