How do I get the window object for a specific tab if I have that tab's tabId?

The window object as seen inside chrome extensions:

 chrome.tabs.get(YOUR_TAB_ID_HERE, function(tab){
      chrome.windows.get(tab.windowId, function(win){ 
           console.log(win); // THIS IS THE WINDOW OBJECT
      });
 });

But if you need the javascript runtime inside a specific tab, you'll need to use Content Scripts which are better explained here:

http://code.google.com/chrome/extensions/content_scripts.html