Suspend this in a small gravitational field, and launch it.
The surface gravity of a neutron star is 100,000,000,000 times greater than Earth's surface gravity. If you can create a field of that intensity, then you don't need neutronium. That also ignores the energy and time required to get the neutronium moving without mass lightening or warp field.
Ah,
Were in the 24th century sir.
(a) Mining would be dome via the transporter.
(b) It would, most likely become a pulsar due to the uneven mass distribution and begin a wobbly spin.
(c) How would using the transporter be difficult? Once transported into an anti-gravity field, everything would be fine from there.
Hey, the 24th century is awesome right? 
The moment the neutronium is out of the star's gravity, the taken neutronium would violently explode. Because, the Strong Force acting on the neutrons are probably too weak to hold them together without gravity.
However, neutronium in Star Trek is naturally stable. The stellar core fragment and neutronium structures show this.
Except that even the late-24th-Century Federation doesn't know how to make anything out of Neutronium. They know OF neutronium materials, but they don't know how to even make neutronium, let alone make anything out of them.
That's actually the good part, you don't need the ability to make anything out of neutronium, you just need a lump of the stuff. If we take not being able to make anything out of meaning they can't make any of it at all, then it's different. I would go with the latter, since I think that makes more sense. It would fit with the idea that not being able to make things really means not being able to make enough neutronium to make anything.
That would place the Federation's ability to manufacture, or otherwise obtain, neutronium at a a level comparable to our current 21st century ability to make antimatter.
Either way, it doesn't really matter what you make your projectile out of, it still takes energy to accelerate. Ultimately, you're only getting out of a kinetic impact weapon the energy you put into it to get it moving. It doesn't matter if you have a projectile made out of a teaspoon of neutronium with three times the mass of Earth, you still have to throw it at the target, and the impact energy is only going to be equal to the energy with which you throw it.
More than likely, photon torpedoes reach high-impulse velocities the same way that shuttles and starships do: with artificial mass reducers. This means that when they impact at high sublight speeds, the kinetic energy is far less than rest mass X 1 / sqrt(1- velocity squared divided by the speed of light squared) times velocity, because the artificially-reduced apparent mass will be far less than the torpedo's actual mass. The total momentum would be the same, and that would be equal to the energy of the acceleration imparted by the launch tube plus the energy of the acceleration imparted by the torpedo's engines.
The existence of artificial mass reducers is probably why we don't see any kinetic weapons in Trek, despite the relatively high sublight speeds we know ships and torpedoes are capable of traveling at - they get going that fast by artificially reducing their mass, so that when they impact they impact with the kinetic energy of a much lighter object traveling at that speed, not the kinetic energy of their non-reduced mass traveling at that speed. There is undoubtedly some kinetic energy involved, but on the scale of Trek warhead and beam weapon yields, it is not likely to be very significant.
Indeed. The only kinetic weapons we ever see are those possessed by primitive civilizations.
The only way around the limitations of the technologies is to use tricks, such as what I previously mentioned. But, that sort of weapon is only good for ambushes, it's not a game changer. It's also a very cumbersome, and probably expensive, method that could be more easily achieved with conventional period technologies.
It's the kind of weapon we might see from a primitive warp civilization trying to fight a much more advanced warp civilization. The primitive society could easily not possess the extreme munitions and drives of the other, so they get around it by the momentum trick. It would be a little like insurgents versus the army, in how they set up ambushes with preset bombs, while the army has organic support and fights where ever.
