It probably has no effect: objects that are not in stable orbits around the earth are going to be in hyperbolic orbits relative to the Earth's gravitational field. A dangerous asteroid would need a periapsis lower than the Earth's radius by a certain amount to hit the Earth, so the risk of collision is still roughly proportional to Area. If gravity is a factor I would expect it to be a high order term. Atmospheric effects are probably more significant.

Interesting question, not sure. As the speed of the astroid is goes up, the relevance of gravity goes down. As the speed goes down, the damage it can do also goes down, and also reduces the probability of an encounter.

The degree of gravitational focusing depends on the speed of the incoming object, but I seem to recall that for typical interplanetary velocities it's of order 1, ie not huge.

It may not change this encounter much, but it could change the orbit just enough for some future earth-orbit crossing - as could the gravitational effect of any of the other planets, their moons, and all the other asteroids change the orbits of anything out there. Solar systems can be messy places, over geologic time frames. Sometime, sooner or later, we will have a significant impact.

And does that make it more likely to miss? Ie it gets slung around by gravity instead of hitting?

Thinking out loud time! My favorite!

To matter for a single pass, the object must be small and/or traveling slowly. Otherwise, itll just keep on cruising.

Iteratively is a different story. Many objects repeatedly cross earths path. How that orbit changes over time i suppose is a function of the other gravity wells the objects pass when they arent near earth. I expect there are so many confounding gravities that only a certain belt\torus exists in the universe where, inside that torus, space stuff is capable of running into us.

Side note: this article is clickbaity because it implies no one knew asteroids almost crash into us without us knowing, but thats not true. People know. There are hard reset buttons floating around in space and we might just press one without warning.

https://en.m.wikipedia.org/wiki/Potentially_hazardous_object