The direction of angular velocity is different from that of regular velocity for (arguably) two reasons. First, it points out of the plane because of the nature of angular velocity. It signifies a rotation, as such, there is not any particular direction unit vector in every coordinate space that could represent it. In spherical or cylindrical coordinates, it would of course be easy to assign it to the $\hat\theta$ direction, but what about systems like Cartesian coordinates? Thus, to signify the direction of something that points in every direction on a plane, it is easy to specify it along the one direction we can be sure the velocity isn't pointing - normal to the plane. This is a much used convention (such as with area vectors, torque, and many others). As usual as well, we use the Right Hand Rule.
The second, and perhaps more important reason is that we always want to ensure that the angular velocity does not correspond to any true velocity that would be moving in a radial direction. However, to convert angular velocity to true velocity, it is necessary to multiply by the radius (for the most part). Therefore, the equation:
$$\vec v=\vec\omega\times\vec r$$
is used. This allows us to define it in such a way that the true velocity never has a radial component due to the angular velocity.
1) Does this mean that for any particle on the rotating body the angular velocity is the same?
On a rigid rotating body, yes, the angular velocity is the same for every point in that body.
2) Does this mean that when angular momentum is described, we are technically still describing a relationship between linear velocity and mass (mv), only now the linear velocity depends on angular velocity and radial distance from axis?
In effect, yes. What you are setting up is an equation of momentum for every infinitesimal mass element of your body. You see the analogy between linear and angular momentum:
$$p = mv$$ and $$L = I\omega$$ where $I$ depends on the distribution of mass, not just on the total mass itself.
3) this would mean that linear velocity would be less for particles close to the axis of rotation, but angular velocity would be the same?
That's exactly what's happening. To visualize this, simply imagine spinning a weight fixed to a string over your head. If you spin one weight with a certain angular speed $\omega$ and then release the string, it will fly off at a certain speed. Do the same now with a shorter string but the same angular speed. The weight will fly off at a slower velocity.
4) Then why would something like a pulsar rotate faster as its matter get closer to the center?
Since angular momentum is conserved, decreasing the moment of inertia increases angular speed: $L = I \omega$. As an analogy, consider a pirouette of an ice skater. If the ice skater has her arms outstretched (big moment of inertia) and rotates at a certain angular speed $\omega$, after she pulls in her arms, she will spin at a faster rate. This is because the angular momentum she had before is the same as afterwards.
All in all, angular speed/momentum follows a neat analogy with linear speed/momentum.
Best Answer
Picture this. You throw a ball into the air while gravity is pulling the ball down, back to Earth. The acceleration of the ball at all times is $$-9.8~\rm\frac{m}{s^2}$$ Assuming that the up direction is positive. The ball will rise until it reaches that certain point, and then fall back down. At the exact moment the ball reaches its highest point, the velocity is zero, but the acceleration is still $$-9.8~\rm\frac{m}{s^2}$$ (Actually, you need a difference in time to compute a velocity but we'll ignore that in this case.) So if the "instantaneous velocity", that is, the velocity at a point in time is zero, the acceleration can be other than zero, but it doesn't have to be. It is also possible that the ball is at rest with no forces acting on it.
In real life, however, this isn't possible. You need to compute the displacement between two separate points in time to compute a velocity. If we assume that there is a force that causes acceleration, the velocity of the two points we measure would be different and thus lead to different displacements.
Angular motion is very similar to linear motion. We can measure displacement, velocity, acceleration, and inertia in both. The relationships between the 4 is the same for both.