I want to slide vertex (01) along edge (02) to be parallel to this vertex(03).
Really Blender dont have slide working with snap?
You need to enter edit mode. Then select vertex which you want to move. And then press G letter twice (Double G). There you go.
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This is simple slide option. I want to slide and perfect align to the vertex no.3
Move your object down slightly, so that your vertice to align to is on that gridline. Then just align your other vertice to that same gridline.
This is one of those things where Blender lacks a concise way to do just that. However given the alignments in this picture, the next best thing is to use the knife tool in orthographic mode with the constraint turned on. That will snap the cut being made with knife to vertical, horizontal, and 45° angles. After making the knife cut, you can slide that vertex up to the vertex made by the new edge and weld it either with remove doubles or by having the auto-weld setting enabled.