After calling chrome.tabs.query, the results are not available

Your problem can be simplified to:

/*1.*/ var fourmTabs = [];
/*2.*/ chrome.tabs.query({}, function(tabs) {
/*3.*/     fourmTabs[0] = tabs[0];
/*4.*/ });
/*5.*/ console.log(fourmTabs[0]);

You expect that the fourmTabs array is updated (by line 3) when line 5 is reached.
That is wrong, because the chrome.tabs.query method is asynchronous.

---

In an attempt to make you understand the significance of the asynchronous aspect, I show a code snippet with the same structure as your code and a story.

/*1.*/ var rope = null;
/*2.*/ requestRope(function(receivedRope) {
/*3.*/     rope = receivedRope;
/*4.*/ });
/*5.*/ grab(rope);

• At line 1, the presence of a rope is announced.
• At lines 2-4, a callback function is created, which ought to be called by the requestRope function.
• At line 5, you’re going to grab the rope via the grab function.

When requestRope is implemented synchronously, there’s no problem:
  You: “Hi, I want a rope. Please throw the rope^{“call the callback function”} when you’ve got one.”
  She: “Sure.” throws rope
  You: Jumps and grabs rope – You manage to get at the other side, alive.

When requestRope is implemented asynchronously, you may have a problem if you treat it as synchronous:
  You: “Please throw a rope at me.”
  She: “Sure. Let’s have a look…”
  You: Jumps and attempts to grab rope Because there’s no rope, you fall and die.
  She: Throws rope _{Too late, of course.}

---

Now you’ve seen the difference between an asynchronously and synchronously implemented function, let’s solve your original question:

var fourmTabs = new Array();
chrome.tabs.query({}, function (tabs) {
    for (var i = 0; i < tabs.length; i++) {
        fourmTabs[i] = tabs[i];
    }
    // Moved code inside the callback handler
    for (var i = 0; i < fourmTabs.length; i++) {
        if (fourmTabs[i] != null)
           window.console.log(fourmTabs[i].url);
        else {
            window.console.log("??" + i);
        }
    }
});
// <moved code inside callback function of chrome.tabs.query>

With breakpoints, your code works, because by the time that the second part of the code is reached, the callback has already been called.