Whenever you copy something, then paste it, it will be pasted at the top of the layer stack in the same level as all of it's sibling layers. Is there a way to make Illustrator paste to the bottom of the stack?
If by "layer stack" you mean "everything on one layer, but I want my object pasted behind everything," there's Edit → Paste In Back.
If you have three layers, and you copy from the top layer and want it pasted on the bottom, in the Layers Palette, make sure "Paste Remembers Layers" is turned off. Copy, click on the layer you want to paste to, and paste.
Quick Keys to learn:
There is no automatic setting for you to paste to "Layer 2" when you're in "Layer 4".
Note: just as you paste an item, look at your Layers palette. That colored square you see represents the selected content of that layer (which in this case is the item you just pasted to the artboard)...drag the colored square to the layer you'd like your newly-pasted element to reside.
I find this method quicker than turning off "Paste remembers layers" <-- this is what I want 99% of the time.