I'm working on a robot for a video game, and some of the bones act different than the rest. Some move different parts of the mesh with unequal force even though the weight is 1.0 on all affected parts. Another problem is were assigned vertices don't move at all even though they are part of the same mesh and weighted at 1.0.
For the shoulder, first select the armature, then shift select the mesh, and switch to Weight Paint mode. Display the right panel with N and choose Blend > Mix, disable Front Faces Only and enable 2D Falloff.
Now select a bone and paint:
For the finger, as it needs a bit more precision, come back to Edit mode, select the vertices, in the Properties panel > Object Data > Vertex Groups, choose the vertex group they are supposed to be part of and click Assign. You can also use this list to remove some vertices from some groups. In the right menu, under + and -, you can sort the vertex groups by name.