I have two panels: Between them is a uneven gap, in some places it is very narrow, in other places it is wide. How can I make it so that the gap is consistent width?
for testing on.
Try this way:
P.S.: For the 3rd step, you can simply CtrlB to bevel those edges. However, the normals for both islands are opposite in your case, you need to unify them before that.
for a simple case like this you can deactivate the Solidify Modifier and use Remove Doubles to connect the 2 pieces.
Now select the edge that is where the gap was and use Ctrl + b to split the edge. Now you can delete the faces to have the gap that was originally there.