How do I bind a key in OSX to open a new window of an application?
There's no single way to open a new window with AppleScript, but make new window or make new document works in many applications.

tell application "Google Chrome"
    make new window
    activate
end tell    
tell application "Finder"
    make new Finder window
    activate
end tell
tell application "Terminal"
    do script ""
    activate
end tell
tell application "TextEdit"
    make new document
    activate
end tell


You can assign shortcuts to scripts with applications like FastScripts or Alfred.

You can also open new windows from the Dock: