You have to do the mirroring/duplicating and application of the scale and rotation to each mesh BEFORE you do any parent linking or animation on ANY arm. Depending on your initial setup of blender, certain things may or may not be copied when you duplicate a mesh.
I'm no expert on Blender and all I can say is what I've done that works for me. The steps I would do are:
Make sure the frame is set to 1 and always at 1 for the rest of the steps
1. Create the right arm - fix the rotation and scale (CTRL+A).
2. Duplicate it and mirror it to become the left arm
3. Apply the scale and rotation (CTRL+A) to the left arm
4. Set the pivots of each arm to their respective base.
5. Create the right hand (not fingers)
6. Duplicate and mirror it (the hand)
7. Apply scale and rotation
8. Fix the pivot point to the wrist of each hand
9 and etc - repeat for each finger.
Then after the models are created, rotation is zeroed and all the pivots are set, now link the parent child relationships.
After the parent child relationships are set, you are now ready to animate.
These are the steps I would follow for the animation:
1. Press A to select all of the meshes (arms, hands and fingers)
2. Make sure the frame is set to 1 and press the I to open the Keyframe setting menu
3. Choose LocRotScale to set the initial pose of all of the limbs to their default position. Press A again to deselect everything
4. Use the arrow keys to increase the frame (maybe to 10 or something)
5. Right click on a limb (the left arm for example)
6. Press R to enter rotate mode. Hold CTRL and move the mouse to rotate the arm in even increments of 5 degrees.
7. When satisfied, press I again and set LocRot .
8. Either keep advancing the frame and setting the animation for this limb, or set the frame back to 1 and Right click on a new limb
9. Advance the frame number for this limb and apply a rotation. Then press I again to set LocRot
10. Keep repeating this process for all of the limbs.
I've attached a simple blender file animation of two rough (just boxes) arms that I used these exact steps on. It's only a 21 frame animation but it shows independent movement of each limb.
Enjoy your day.