Why there is current in a neutral conductor, but no voltage?
Well, actually, when we say that the voltage at a point X is equal to a certain value, what we're really doing is measuring the voltage difference between this point X and an implicit point. In our case, this implicit point is called the ground - in the electrical grid, neutral is the ground by our choice, with voltage equal to 0.
Now, ignoring our arbitrary choices and assuming that our neutral wire is a normal conductor, then if you measure the voltage between two different points in this neutral conductor, which has some current flowing in it, then you will in fact find a small difference!
In most cases, this voltage difference is so small that it can be ignored for most practical uses. If this voltage happens to be big, then your neutral conductor is not doing its job.
An interesting fact is that this is true only because our neutral conductor is assumed to be a normal conductor and not a superconductor. If that were the case, then you could have current flowing in it, but no voltage difference between any two points in it! In other worlds, current can keep on existing when you have no resistances in your conductor, even when the voltage difference is 0 anywhere on it! Voltage would be necessary to keep the current flowing only when you have an internal resistance.