As the fluid travels into the pipe, the effect of viscous forces are felt further into the fluid, from the outside, i.e. pipe/channel wall.
In laminar flow, Re < ~2000, there will be a parabolic velocity profile. Turbulent flow has a different profile, but it is more or less similar the further downstream once the flow is 'fully developed'.
Inertial referes to the kinetic energy and momentum of the flow, and the pressure parallel with the flow. The viscous effects refers to the friction and shear forces on the flow.
For incompressible flow. It has to do with conservation of mass (continuity equation), not viscous effect.
I'm not sure about v=0, w=0 unless that refers to y, z directions, and u refers to velocity in x. The fluid is contrained by the channel/pipe in one or 2 dimensions. Wide and long could be a river, for example, so there is a characteristic velocity profile which is a function of depth only (ignoring the channel profile and banks). But basically, for a very wide channel, the velocity profile is a function of depth only (hence 1-D), and the profile is much the same far downstream, assuming the no change in the elevation or the channel geometry, i.e. uniform depth and channel profile, across and along the channel.