"So, the 2nd law of thermodynamics means that everything tends towards high entropy. All the stars will burn out, but where does all that energy go?"
-Order and disorder have to be defined first, you could say that a soup of atoms is order, or disorder and life is order or disorder, the energy will be wasted into space, turned into entropy or mass, or arguably alter a different universe, balancing out the law of conservation of energy.
"They burn out but the heat they emit must still be around, no? Does the radiation just dissipate into nothing?"
-It just turns into mass, unusable energy or 'disorganization' like entropy.
"What about stars that supernova? Don't they turn into nebulas, that can then re-form into stars?
Or will all the matter be inside black holes? Don't the black holes evaporate? Where does the energy go when they evaporate?"
-Remember, energy is never created nor destroyed, and we dont exactly know where the mass in a black hole goes, my theory is it becomes inaccessible through a hole in space-time, exactly like the klien bottle (http://upload.wikimedia.org/wikipedia/commons/2/21/Acme_klein_bottle.jpg
-Now also remember that energy can be transferred into multiple types of energy, potential in the mass of objects, gravity, mass, light, heat, whatever you please, so therefore the unaccounted for energy is always there though we might not be able to trace it directly.
"Does the fact that the universe keeps expanding mean that the energy will just get so spread out that it will be stuck in the highest state of entropy? If the universe didn't expand, would the universe never die out?"
-There are many theories on this matter, such as the big tear, the big crunch etc. (look those up if you please)
"I assume this will happen, so does this mean that there will be no such thing as time when it does happen?"
-If the big tear were to happen, yes time would become useless.
"Would it then be possible for particles to spontaneously come into existence from this vacuum and create another big bang"
-Yes, if that theory is correct thats a possibility, we might never know though.
-I suggest you read Brian Green's books, The fabric of the cosmos, The hidden reality, and the elegant universe, as they would help you and you would love to read them! If you hate reading there are also DVD's on the matter and i will put links to all this stuff below for you and anyone else!
Hope this helps feel free to ask any more clarifying questions!