Is there a way to make a window in OS X Lion always initially appear on a select screen? So that when the window appears, it'd automatically appear on the screen I select?
If by screen, you mean desktop, I think this is how you can do it:
Go to the desktop you want.
On the doc, right click (or control-click) the item you want to appear on a certain screen. There should be a menu with "options" on it. Select options, and from the submenu select "this desktop".
Return to any other desktop and verify, that if you were on, say 'desktop 2' when you selected 'this desktop', the options will have 'desktop 2' checked.
Then if you click the app on the dock from whatever desktop, your desktop will switch automatically. Results may not be the same when you open the app with some different method. For example, when I tried from a terminal window "open -a Bean" it opened Bean in the right desktop, but did not swithch automatically. When I tried to open Firefox with open -a, basically nothing happened (apparently).
(I'm on Lion 10.7.4, but as I recall this also worked in a similar way on earlier distributions.)