I can kind of give an answer, but I welcome correction.
I was wondering how a pulsar would appear to a human being, in visible
It wouldn't look like much in the visible light spectrum unless there was a significant nebula, then we might see the effect of the pulsar on the nebula, but not the pulsar itself. X-rays and radio waves aren't visible, and if the pulsar wasn't directed at us, we wouldn't see it pass through empty space.
Neutron Stars are generally too hot for us to see. If one was to cool down significantly, to maybe 10 or 20 thousand degrees on the surface, then it might glow visibly blue and look like the brightest star in the sky, still just a point in the sky, but the brighest point in the sky at 1 AU.
But mostly they're too hot to glow in visible light.
What you might see from 1 AU from a Neutron Star could be the accretion disk. Matter that falls into a Neutron Star gets very hot and the energy if impact is far greater than the energy of fission, so as matter gets close to the Neutron star and spirals in, you're probably talking x-rays and gamma rays, but you might see a visibly glowing accretion disk at some distance out, perhaps in a gradually decaying orbit. In effect, what you could see would depend on what's around the Neutron star than it would depend on the star itself.
As I understand, the pulsar's beam is projected from the star's
magnetic poles rather than rotational poles, which are not necessarily
in line with each other. Given that pulsars rotate extremely quickly
and the beam could be visible across vast distances - such as if it
were shining through the pulsar's nebula - would it appear as a
straight line, curved line or perhaps a cone
The problem here is, you can't see the beam. You see light as it's pointed towards you, you can't see a light beam in space (even if it's visible light).
You can see a beam not pointed at you in the atmosphere because of reflection off dust and water molecules in the air.
(see little picture)
In space, matter is far more spread out. It's true that a pulsar can light up part of a nebula, though the nebula may also glow on it's own anyway (I'm not 100% sure on that), but a Nebula is very large and very spread out. To see it from the naked eye, I don't think you'd see much other than perhaps a large glow.
If you could see a pulsar beam, it takes light 8 minutes to for light to travel 1 AU, and a pulsar can rotate hundreds of times, perhaps thousands of times in 8 minutes, so if you could actually see the beam, it would be enormously curved, like a spiral. The light itself would travel in a straight line but since the source of the light was rapidly rotating it would appear like this (picture below), if there was sufficient material for the light to reflect off of (which there probably wouldn't be, not within 1 AU).
In reality, it would look nothing like that, but if you could see the the beam, that's what it would look like. What that spiral looks like from a single point is a pulsar, off, on, off, on, off, on, etc.
Also, the light never travels in a spiral, it travels in a direct line away from the Pulsar, but like the water spiral here, which falls down in a straight line, but it looks like it falls in a spiral (if that makes sense).
Given the incredible density of neutron stars and their small physical
sizes, would the night sky be visibly distorted to the point where
(for example) just after sunset on a hypothetical planet, one could
possibly observe other planets near or behind the star that would
otherwise be blocked by it?
Well, for starters, without a sun there, planets would probably not be visible. If the Neutron Star glowed brightly due to a hot accretion disk you couldn't see anything behind it cause the brightness of it would make seeing light bent around it pale by comparison.
Now if the Neutron star was dark, to our eyes, then we could see gravity lensing around it, but stars, not planets cause planets would be dark. (The moon would be very dark too, visible more by what it blocks than what it shines). The lensing would be quite small however. Visible lensing would only be a few times the diameter of the Neutron star, maybe 100 miles across, which, 93 million miles away is really tiny. You might see some odd warping of a star here or there when properly lined up, but to see any interesting visible lensing you'd need a pretty powerful telescope.
Given their small surface areas, would a neutron star still appear as
luminous as say, the Sun, at a similar distance? How close would you
have to get to a neutron star for its apparent magnitude to match the
Sun's from Earth?
Kind of touched on this above. The Neutron Star can give off a lot of energy in it's pulsar beam, but it's mostly x-rays, not visible light. How bright it is would depend on how much material is falling into it at the time, so there's no right answer to how close the Earth would need to be to have equal brightness. It's a different kind of brightness too, mostly not visible light. But there's no way to answer that question cause it depends on too many things.
When a Neutron star is just formed (which usually happens after a supernova so there's enormous energy released), but when the star just forms, it's maybe 12-15 miles in diameter but it's surface temperature can be (guessing) perhaps a billion degrees, though it cools very quickly. A very young Neutron Star might emit more energy to our sun, though much of it would be in Neutrinos that would largely pass through the Earth. But that level of energy output wouldn't last long. It would cool down to about a million degrees within a few years. Source.