Yes, your IK’s are lagging. A normal IK arm is set with the arm being influenced by the Hand IK. You are correct to have a “Copy_Rotation” for the normal hand to the IK.
For the arms you will need two extra bones called Pole Targets. These Targets will allow you to position the arm correctly. Depending on the the bone rolls you might get some odd results, if so adjust the pole angle. Usually the pole angle can be adjusted with -90, 90, or 180 degrees should work.
Naming convention for blender bones is .L for left and .R for Right [Left Arm Lower = Arm Lower.L]
Conclusion:
+Hands Should only need a “Copy Rotation” Modifier to the IK bone.
+Arm Lower’s would need the “IK” Modifier instead of the hand.
–Pole Target bones should be made if using IK’s
–Adjust Pole Angle to keep arms perpendicular to the ground
—Pole targets are location bones that points where the elbow bends to.
+Names for bones could be changed with .L’s or .R’s for faster animation.
–If using this in a game I would recommend using underscores (_) instead of spaces. This will help with scripting.
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GAME RELATED:
It sounds like you are trying to build this character for a game engine. Not sure if you are but just incase please read on. Deform Ticks should be off for bones you do not want in the engine. To find this deform tick box you need to go to your Bone tab. It’s right beside the Bone Constraints tab. Scroll down and you will see “Deform” with a tick box you can check or uncheck. Now that you have unchecked the bones you didn’t need like (pole_targets and IK bones). You can now export the armatures under FBX. I would recommend using the 6.1 ASCII. The 7.4 binary gives odd results but you can try though. It’s just within my experience that it gave an extended name to each of the bones. I have never tried a multiple armature fbx so I’m not sure how it will go. Be sure to select the objects and armatures you would want for the single FBX before you export. Once here you should get a export page and a settings down on the lower left of your screen. This is where you have your export settings. After export it does a natural bake to those Deform bones for both 6.1 and 7.4 either you choose.
The following is what you might need to change. Anything else from these settings will give alternate results. What I don’t say you shouldn’t need to change. Apply Modifiers for instance, you’ll want this checked of course.
Version FBX 6.1 ASCII (Export Settings):
Selected Objects [Checked]
Forward:(Depends on your engine) Unity/-Z Forward
Up:(Depends on your engine)Unity/Y Up
[Armat][Mesh] #The only things checked/highlighted
Checked Tick Boxes:
Only Deform Bones
Animation
All Actions
Optimize Keyframes
Unchecked Tick Boxes:
Loose Edges
Tangent Space
Default Take
Version FBX 7.4 Binary (Export Settings):
Selected Objects [Checked]
Forward:(Depends on your engine) Unity/-Z Forward
Up:(Depends on your engine)Unity/Y Up
[Armat][Mesh] #The only things checked/highlighted
Checked Tick Boxes:
Only Deform Bones
Baked Animation
Key All Bones
All Actions
Unchecked Tick Boxes:
Loose Edges
Tangent Space
Custom Properties
Add Leaf Bones #If you want to edit this armature from export. I personally don’t use them.
NLA Strips #I haven’t found a use for it but if you do you can “Tick” it