"2. Whenever I shoot a bullet it erases my previous bullet when I press the shoot key. How can I make it so multiple bullets are on screen at the same time? "
What did you do to create the bullet?
"4. This is related to the third issue, but how do I kill my character and show the death animation? Right now the death animation shows only when I am standing still. Plus I can still move around for about a second after I die."
you can make a instance variable (click on your character on the layout view, and to the proprieties menu (to the left by default), you should see instance variables) that will be used to know if you are alive or dead, if you are alive, you will be able to move, else, you will not be able to.
And you do the same for the ennemy, and makes so he needs to be alive to kill you.