Would a magnetic needle standing on it's point on a magnet be more stable than the same needle stood on a non-magnetic surface? So I was watching MinutePhysics' video 'How long can you balance a pencil?' and it turned out to not be for very long. 
Would introducing a magnet to the equation make for different results?
We can assume the magnet and non-magnetic surface are both uniform and perfectly flat, and that the needle is uniform to a 'sharp' tip (sharp to a point where material compress-ability is not a factor, so it would technically have a microscopic flat surface).
I think it would help due to the magnet securing the flat portion of the needle against the magnet evenly, but then again, with a slight offset, it would act on the shaft of the needle with any amount of horizontal propagation...
 A: This, to me, depends on how uniform a magnetic field your magnet creates in reality.  I am no expert, but if it's not, I can imagine you are doubling your chances of the needle falling, both from the unevenness of the surface and the possible unevenness of the magnetic field.
Sorry for the short answer, but I hope you get a better one. 
A: I'll assume that you're aware that all of this is described by differential equations. For instance, here's the equations for the inverted pendulum. What you seem to be interested in are the cases where the angle $\theta$ remains a constant. 
In the case of the needle on a flat surface, no magnetism, there are two steady states. 
A) The needle stays upright forever
B) The needle falls down and then stays on its side forever
You can classify these "equilibria" by linearizing the differential equation about the point. Look here for a good introduction.
The fixed point in A is an unstable fixed point. However, the point in B is stable. This means that slight variations in the angle $\theta$ that allows the needle to stay in A will force it to go to B. 
I'd think of it like this. There is only one angle that allows a needle to stay upright. Namely, $\theta={{\pi} \over 2}$. However, there are an infinite number of other points that allow the steady state angle to become $0$ or $\pi$. So even a microscopic change in $\theta$ from ${{\pi} \over 2}$ to ${{\pi} \over 2} +\epsilon$ will move the angle from a steady state angle for A to a steady point angle for B.
What happens when there is a magnet? The magnet will pull in the same direction as gravity. The difference will be the strength at which the magnetic force pulls. At the bottom of the needle, the force will be quite strong, while the force at the top will be weaker. In fact, the magnet will "fix" one end of the needle about the surface of the magnet! This will quite literally make it an inverted pendulum. Therefore, all the above analysis applies.
