If the window's white plastic, then its prolly a vinyl replacement window. Some of those tilt in for cleaning. So you could put the mesh on the outside and tilt them in for cooling. The chickens might roost on the tilted in sash though; making a mess on the glass. You might put 2-faced tape on...
Here's a rough drawing I made:
http://i59.photobucket.com/albums/g314/tomd7735/Coopwindow.jpg
Maybe you could drill a row of 1" holes at the top of the wall under the roof overhang or eaves; to get a more even draft instead of a wind tunnel with the one hole.
Tom
Is this double hung window an older wooden one? It is possible to convert it to a double awning type by removing the exterior window track stops and putting in 4 pins through the outer frame into the top of each movable window. So they both can swing out at the bottom, the upper one swinging...
Thickness wouldn't matter much as long as you build it tight to reduce drafts. With plenty of straw or shavings the peeps can stay warm enough, but through March and April you might need to hang a heat lamp from the ceiling overnight. You might point it at the waterer to keep that from freezing.
Tom