It would all depend on how close the neutron star was. If it was outside the event horizon, then if would be observed to be orbiting "nothing".
If it strayed too close to the black hole, then it would be slowly ripped apart, until a slightly larger black hole was all that is left.
Convert into a dwarf white star which is really hot, then a neutron star, and finally, a black hole.
A black hole has more mass than a neutron star, but if you are comparing volume it would depend on the mass of the black hole. A neutron star is estimated to be about 14 miles in diameter, which is larger than the event horizon of a black hole up to about 3.8 times the mass of the sun. A more massive black hole will be larger.
No, black holes cannot turn into neutron stars. Neutron stars form from the remnants of supernova explosions of massive stars, while black holes are formed from the gravitational collapse of massive stars. Once a black hole is formed, it will remain a black hole and will not transform into a neutron star.
In all probability - not that this scenario would happen - but the resulting combination of masses, would push the combined "stars" over the Chandrasekhar limit and a black hole would form.
If you jumped into a black hole, you would be stretched into human spaghetti.
A black hole or a neutron star.
nothing
The black hole's mass would increase by an insignificant amount.
You can't just destroy it - it would take a huge amount of energy to tear it apart. About the only way I can think of to "destroy" it - in away - is that it collides with a black hole, and the mass of the neutron star becoming part of the black hole - or the neutron star itself becoming a black hole, if its mass increases (due to additional mass falling into the neutron star).
A neutron star or a pulsar, or a black hole.
You would die
Adding more mass to a 1.4-solar-mass neutron star could potentially push it beyond the limits of neutron degeneracy pressure, causing it to collapse further. This could result in the formation of a black hole if the mass exceeds the Tolman-Oppenheimer-Volkoff (TOV) limit for neutron stars.