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.