If the music is on your computer, you can import it into iTunes and sync it on your phone, even if the music was not originally from iTunes.
Put it in a bag of rice gently and take it to an iPhone repair store. But you may have buy a new one.
You can use the same SIM card for your iPhone 6 that you used in your iPhone 5.
The animal in the Sprint iPhone 6 commercial is a dog. More specifically, it is a hound dog.
The voicmails are not on your phone - they are on the computer server of your network provider. You are already aware of how to listen to the messages. Whether or not you can get them downloaded from the network would depend on your provider !
One long-winded possibility would be to play them back while your mobile is on 'speaker-phone' - and record them to your computer via a microphone.