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.
In general, yes. If different parts of the body have different angular velocities (or different angular accelerations, which will result in different angular velocities), then you have particles moving relative to one another. Assuming an object is a rigid body means that all particles are fixed in place relative to the body. If the body is not rigid, then this allows for relative motion, and hence different angular velocities.
Of course, you could always take a non-rigid body and move it in such a way as to make all parts of it have the same angular velocity or angular acceleration. But this is a contrived case.
Best Answer
Consider a rod held vertically with a pivot at its base. If the rod is allowed to fall to one side each point on the rod turns through the same angle in a given amount of time. This is shown in the diagram bellow:
To say otherwise would be against all experience. The definition of angular acceleration is the rate of change of angular velocity or the second derivative of angle with respect to time i.e. $\frac{d^2\theta}{dt^2}$. But since for all points the angle they change is the same in a given amount of time their angular velocity must also be the same and hence the angular acceleration for each point on the rod is also the same.
Now to linear velocity. The distance travailed by any given point is given by $x=\theta r$ (this is simply arc length) where r is the distance of that point from the axis of rotation in this case the pivot. Linear velocity is the derivative of this with respect to time: $\frac{dx}{dt}=r\frac{d\theta}{dt}$. Since this depends on the distance of the point from the axis of rotation ($r$) which increases as you move along the rod away from the pivot this is not the same for each point on the rod. Similarly linear acceleration is given by $\frac{d^2x}{dt^2}=r\frac{d^2\theta}{dt^2}$ and again it to depends on distance from the axis of rotation and thus is not the same for every point on the rod.