Obviously the inside of a black hole does exist on account of the fact that it is able to warp spacetime in an externally detectable manner. If the interiors of black holes didn't exist they would be indistinguishable from empty space, and not seem so remarkable as to be called "black holes."
Either way, we do not know what processes occur inside because we lack any theory which could describe it. This certainly doesn't mean nothing happens in there or nothing exists in there.
>So far as our universe is concerned, the inside of a black hole literally doesn't exist and events in there literally don't happen.
It would probably be easier to get this across if I substituted "So far as our universe is concerned" with "From our perspective outside of the event horizon". Remember that a singularity doesn't just "pull" light so hard that it can't escape - they warp spacetime around them, so much so that the space and time within the event horizon ceases to be meaningful from our reference frame. The only way to go is down - literally. If you turn 360 degrees in a black hole you will never face the outside.
Also, we definitely have theories that describe what could happen inside an event horizon. General relativity, for example. It's from these that we can establish that space and time "switch" within, for example. We can solve general relativity for the conditions within an event horizon and make some pretty strong predictions about it (it's from these predictions that I conclude elsewhre in this discussion that fusion is unlikely).
The inside of a black hole doesn’t exist as far as our universe is concerned. Even the ‘gravity’ isn’t ‘emanated’ by the mass ‘inside’ the event horizon; rather it is a ‘recursively’ generated field (remember: gravity is generated by mass-energy curving space-time, gravity is a form of energy, and so gravity begets gravity in an entirely self-sustaining manner).
Either way, we do not know what processes occur inside because we lack any theory which could describe it. This certainly doesn't mean nothing happens in there or nothing exists in there.