My theory is this: (please note I am only 16 and dont have a lot of time to look into these sorts of things in detail, but this is my proposition.)
Energy creates matter through the creation of quarks, leptons, etc, because energy cannot be created or destroyed, therefore had to have been expelled from singularity at the point of the big bang due to approaching an increadibly large density contained within an increadly small area. The expulsion of this energy would have caused heat (the transfer of energy) and when the conditions were right, gamma rays, x-rays, etc form, then neutrinos and photons then up down charm positive and negative energy forms the quarks, leptons, which form the protons and neutrons of atoms, etc. So to answer your question, I think it always has existed in the form of energy.
16 years old!!?? I hope you're out of high school and in University.
Here's one issue I can see with your statement about fusion creating the heaviest possible elements. There are Neutron stars. They're less dense than black holes and more dense than atomic matter.
What happens is that a star collapses to the point where gravity overcomes the electromagnetic force that keeps electrons orbiting around a nucleus. The electrons get crushed into the protons, forming more neutrons.
What you end up with is wall-to-wall neutrons, essentially a HUGE nucleus. Neutron stars are that dense. Visualize an atom the size of a football stadium. The nucleus will be smaller than a fly. Now visualize matter made up of nothing but nuclei packed in until they're in contact. Neutron stars are that freaking dense. They're way beyond your 'heaviest possible elements' scenario, and fusion has stopped, kind of by definition, when you hit a neutron star's density.
Then, when the neutrons collect in a sufficiently large mass, their gravitational force crushes the neutrons themselves, causing a black hole.
It's an interesting theory that could be explored. As I am only 16 I have not had too much time to think about these things, however I grow fascinated by the day. One thing in particular that fascinates me with the universe is that it is so complexing, that it is actually quite simple.
What you're thinking of is along the lines of the Big Crunch then a Big Bounce. I know you stated that a black hole could explode causing a big bang, and that's worth looking into, but essentially if the mass of the universe exceeds the gravtiational pull of the universe (or something along those lines) the universe will contract on itself until it reaches near a point of singularity. Then, the density and size would be incredibly small as Park Bierbower stated, "the size of one atom", and the forces that are contained are too great, causing another Big Bang.
Thus, it could be theorized that the universe is caught in an infinite loop of bounces, "banging" and "crunching" over and over again. Each bounce will have different norms. I was reading an article about parallel universes existing, and it basically stated that during the incredibly small amount of time after the big bang, if the first proton is made with different amounts of quarks/leptons, etc then the universe structure would be different, etc.
So essentially, we are the product of infinite chance, and the reason we see ourselves as perfect is because it's what we've accepted for so long.
Their question belies fundamental flaws in their argument. Firstly, where did God come from if everything has to be created? They have no answer. Secondly, just because we don't know yet, doesn't mean that there isn't an answer. In fact, it's what separates us from the 'faithful'. We are willing to be unsure until there is verifiable evidence. Thirdly, absence of an answer is not the proof of a different answer. This is a common fallacy called 'argumentum ex silentio'. No matter how you look at it, they have no valid argument.
For what it's worth (I am not a scientist), as there is not a single example of an actual beginning in the known universe (not a transition, evolution or conversion but an actual creation) then I assume, until proven otherwise, that the universe itself has always been here. The radiation and expansion that we measure could easily be the result of a 'perfect storm' with previous material exploding in a 'touch paper' fashion that had a knock-on effect, leading to the universe as we see it today. This is just conjecture but it does fit with what we see.
I hope this helps.
I think a better question would be:
Where did the underlying physics apparently in place before the big bang come from?