Some edges don't show in object/weight painting mode

And I think this is not letting me weight paint some vertices. Why does this happens?



I think this are two different things.
If you wanna see all Edges in Objectmode, go in EditMode(logically :-)) and in the More Panel choose All Edges.
To do some WeightPainting you should create a VertexGroup first.
And if Painting still not working, perhaps the weight in the Paintpanel is 0 (blue).