1. Why boundary layer will increase when the flow go along the pipe?

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.
2. When the viscous force take the dominance, what will happen?

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'.
3. Actually I don't really catch the meanings and can't visualize what "inertia effect" and "viscous effects" are.

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.
3. I just read a book which says "the flow is fully developed, i.e. the velocity profile is constant along the direction on flow" is it true? And that's a consequence of the dominance of the viscous effect?

For incompressible flow. It has to do with conservation of mass (continuity equation), not viscous effect.
4. I just read a question in which the following assumptions are made:
"The open channle is wide and long such that the flow is fully developed"
And then the solution says by this assumption, the velocity compoent v =0 and also w=o. Therefore this is just a 1D flow.
Can anyone explain me why?

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 1D), 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.