Is the following correct: when a black hole forms, spacetime encurls around at the black hole event horizon such that spacetime does not exist inside the event horizon.

What do you mean by "encurls"? False; the event horizon isn't a physical barrier: particles cross it all the time without problems. So obviously, spacetime still exists within the event horizon.

The various theoretical solutions to Einstein's equations describe (at least) two distinct spacetime regions, inside and outside the event horizon of a black hole. The solutions tell us that the spacetime inside the horizon is unlike the spacetime we are familiar with, but that doesn't mean there is no spacetime there. Of course, there's no way of directly probing inside a black hole. We can only ever deduce what it's doing by looking at the outside. What we do know is that the outside spacetime conforms to the solution of the equations. There's no reason to suppose that the inside spacetime wouldn't similarly conform to the derived interior solution (within the known limits of applicability of general relativity).

Then I'm pretty sure the answer to the question posed in the OP is that it's not correct. Time doesn't become circular outside the black hole event horizon, so clearly spacetime doesn't fold into a sphere. (Also, not all black holes and/or event horizons are spherical.)

No, a black hole is more accurately represented as a drain pipe... in fact, black holes swallow spacetime easier than supermassive black hole can eat matter. Because the black hole spins very fact at supermassive scales, its pretty difficult for matter to fall in, though it does. But most of it gets flung out of the poles of the black hole. A black hole drags space towards it and it will be swallowed by it.