2nd option is they enter into orbit around each other. Pretty much the same with any 2 massive objects passing near each other.

With classic (non Kerr) black holes, if the Event Horizons touch the black holes have no option but to merge. The Event Horizon is directly proportional to the radius, so when the 2 event Horizons touch it, as that happens, no longer 2 black holes but 1 black hole. The mass of the 2 objects combined has an event horizon that circles both of them.

For Kerr black holes this question is more complicated and I'm not sure but I think it's the same. once the event horizons touch it's no longer 2 separate black holes but one bigger one.

When you're talking about Event Horizons, Gravity always wins. Kinetic energy cannot exceed the speed of light, so it loses.

A long string doesn't make any sense. A temporarily non-round event horizon as the holes merge seems possible, but think of a merger of 2 stretched balls, not a string. Each black hole is centered around it's "singularity", those shapes might warp some as the holes collide or spiral into each other. As an FYI, direct collisions would be rare. Spiraling into each other is much more common.

A few points to consider regarding a black hole. Anything that falls within a black hole's photon-sphere is unlikely to ever escape, so one of the magical distances to consider is 1.5 times the radius of the black hole where all orbits are destined to fall into the black hole. If the Black hole and Neutron Star are of similar size and the Neutron Star passes close to the event horizon and relativistic speed, then there's an interesting scenario where some of the Neutron Star (at close to the speed of light) would be outside the photon-sphere and have the momentum to escape and some of it would not. Neutron Stars are bound together very tightly, but the tidal forces would be very strong and it's possible that some of the Neutron Star material could get torn off in this hypothetical scenario. Would the entire neutron star be torn apart? Maybe. It's hard to say. The Math in that hypothetical is way above my pay-grade.

If the Neutron star is much smaller than the black hole, then it's likely inside the photon-sphere and it will spiral into the black hole, perhaps breaking apart in the process into an accretion disk, perhaps not, but it will get absorbed into the black hole if it's entirely within the photon-sphere, even at nearly the speed of light.

The problem with this hypothetical is that it's pretty much impossible. Objects the size of stars don't accelerate at nearly the speed of light relative to other near by stellar mass objects. What actually happens is described in the first answer to your question, the two dense objects would either have sufficient relative velocity to fly past each other or they would get caught into a mutual orbit that would likely lead to a merger. In the actual universe, a close to the event horizon fly-by between a Neutron Star and a black hole would result in a merger of the two. In theory, with enough kinetic energy the Neutron Star could fly close to and still fly past the black hole, but that much relative velocity probably never happens.

Second option is they enter into orbit around each other. Pretty much the same with any two massive objects passing near each other.

With classic (non Kerr) black holes, if the event horizons touch the black holes have no option but to merge. The event horizon is directly proportional to the radius, so when the two event horizons touch it, as that happens, no longer two black holes but one black hole. The mass of the two objects combined has an event horizon that circles both of them.

For Kerr black holes this question is more complicated and I'm not sure but I think it's the same. Once the event horizons touch it's no longer two separate black holes but one bigger one.

When you're talking about event horizons, gravity always wins. Kinetic energy cannot exceed the speed of light, so it loses.

A long string doesn't make any sense. A temporarily non-round event horizon as the holes merge seems possible, but think of a merger of two stretched balls, not a string. Each black hole is centered around it's "singularity", those shapes might warp some as the holes collide or spiral into each other. As an FYI, direct collisions would be rare. Spiraling into each other is much more common.

A few points to consider regarding a black hole. Anything that falls within a black hole's photon-sphere is unlikely to ever escape, so one of the magical distances to consider is 1.5 times the radius of the black hole where all orbits are destined to fall into the black hole. If the black hole and neutron star are of similar size and the neutron star passes close to the event horizon and relativistic speed, then there's an interesting scenario where some of the neutron star (at close to the speed of light) would be outside the photon-sphere and have the momentum to escape and some of it would not. Neutron stars are bound together very tightly, but the tidal forces would be very strong and it's possible that some of the neutron star material could get torn off in this hypothetical scenario. Would the entire neutron star be torn apart? Maybe. It's hard to say. The math in that hypothetical is way above my pay-grade.

If the neutron star is much smaller than the black hole, then it's likely inside the photon-sphere and it will spiral into the black hole, perhaps breaking apart in the process into an accretion disk, perhaps not, but it will get absorbed into the black hole if it's entirely within the photon-sphere, even at nearly the speed of light.

The problem with this hypothetical is that it's pretty much impossible. Objects the size of stars don't accelerate at nearly the speed of light relative to other near by stellar mass objects. What actually happens is described in the first answer to your question, the two dense objects would either have sufficient relative velocity to fly past each other or they would get caught into a mutual orbit that would likely lead to a merger. In the actual universe, a close to the event horizon fly-by between a neutron star and a black hole would result in a merger of the two. In theory, with enough kinetic energy the neutron star could fly close to and still fly past the black hole, but that much relative velocity probably never happens.