Q:

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

I have the tabId of a tab.  How do I get it's window object? 

A:

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

A:

To get the DOM window object from a tabId, you should insert a content script in that tab:
chrome.tabs.executeScript(tabId, {code:'var w = window; console.log(w);'});

https://developer.chrome.com/extensions/tabs#method-executeScript
Perhaps you'll need to comunicate with your background page:
https://developer.chrome.com/extensions/content_scripts#host-page-communication

