If the center of a black hole is an one-dimensional point, gravitational singularity, where density and gravity become infinite and space-time curves infinitely, then / why:
If spacetime is warped almost infinitely, why isn't it warped to the point where it would pinch off the singularity from regular space-time? (basically pinched off in its own spacetime)
How does a 1 dimensional item affect 4 dimensional space?
It's not really true that a black hole singularity is one-dimensional. Actually general relativity doesn't provide a well-defined answer to the question of how many dimensions it has. See Is a black hole singularity a single point? .
If spacetime is warped almost infinitely, why isn't it warped to the point where it would pinch off the singularity from regular space-time? (basically pinched off in its own spacetime)
If you want to learn about this kind of thing, the phrase to google on is "topology change." General relativity predicts that under a certain set of reasonable assumptions, topology change never occurs. Specifically, it predicts that there is no topology change when matter undergoes gravitational collapse to form a black hole.
We don't actually have any empirical data on the kind of extreme spacetime curvatures that occur when a singularity forms, so it's hard to know whether we should believe GR on this prediction or not. Probably the only way we would get empirical evidence would be if gravitational collapse can form naked singularities, violating the cosmic censorship hypothesis. Some reputable relativists actually think this is a serious possibility for real-world gravitational collapse of stars, but it's not a majority opinion.
How does a 1 dimensional item affect 4 dimensional space?
Well, we can't really say it's one-dimensional, but anyway it's not valid to think of the singularity as affecting the surrounding spacetime. The singularity lies in the future light cone of all observers, so it can't cause anything to happen. For a real-world black hole that forms by gravitational collapse, the effect on the outside universe is basically just the gravitational field of whatever matter formed the black hole originally.
A black hole is a four dimensional hyperspherical object tucked away from our view in thee dimensional space. So basically, from our point of view its not one but zero dimensional. Here's my best analogy: take a circle (which is a sphere in three dimensions) and draw it on a sheet of plastic. Then turn the sheet on its side and now the circle is a one dimensional line. Now compress the sheet's length to zero to match the one dimension of the line (which compresses the circle drawn on the sheet to a true line). Now take it one more step and turn the one dimensional sheet ninty degrees so that you are looking at the line drawn on it head on and it's now a zero dimensional point. This is what makes a black hole a point like object. This is basically why when looking at a black hole that we can see what's behind it on its edges. This is also why to my understanding that particles with zero mass seem point like as well but have wave like properties.
