Frustrations on the order that layers appear when copy/pasting in After Effects are usually apparent in forums. However, the remedy is quite simple. Select the layers in the order you want them to appear when pasted.
If you want layer 1 to be on top, layer 2 in the middle and layer 3 on the bottom, then select layer 1 first and layer 3 last. Then, copy and paste into a new comp and they will appear in that order.
Watch the short After Effecs tutorial below on what happens when you copy/paste after selecting from bottom to top, top to bottom or random layers.
