I understand we must group the layers to then make a movement as a group.
But what if you only need to move part of each (lines & fill)?

You can drag a layer out of a group, to anywhere else on the layer stack. Is that what you mean by "move part of each (lines & fill)"?

Not quite like that.
Imagine you have A and B in the same frame.
Except the outlines appear in one layer, and the color fill in another.
Now, you want to move only B from its location to 1 inch further. But you want both the outlines and the color fill to move together. Is there a way to do a special kind of selection that will go through from one layer to the other, or something similar?

If you select the group layer and then draw a selection outline around the object 'B', you can use the Transform tool (next to the Move tool in the Toolbox). It defaults to 'Position' mode and then behaves as a more sophisticated moving tool. This will move the contents within the selection for all layers in the group.

However, it may be better in the long run to have objects A and B in different groups to give greater flexibility and freedom from interference with each other during various operations.

P.S. If you have them in separate groups, you can then use the Move tool with the 'Move the whole group' option set to move every layer in that group, which only contains one 'object'.