Is it possible to have PHP print in the while loop before it finishes processing?

Although it's possible by controlling the output buffer, I wouldn't do that, mainly because it will delay the JavaScript DOMReady event.

If you're looking for a visual effect, you should use JavaScript (but I don't see any reason for Ajax based on what your question says). It can be accomplished with a simple setInterval. Considering an initially empty <div id="hello">:

var i = 0;
var timer = setInterval(function(){
    if(i == 10000) {
        clearInterval(timer);
        return;
    }
    var div = document.getElementById('hello');
    div.innerHTML += 'hello ' + i + '<br>';
    i++;
}, 250);

I just saw your edit, and now I think you actually should use Ajax! The example you linked to does. Basically, you have to setup a test queue in JavaScript, where each test has an unique URL. Then it's just a matter of firing one request at a time with Ajax (jQuery and $.get would be the easiest way to go). The following assumes an <ul> instead of the div from my previous example, and that the server will respond with a success or failure message for each test:

var tests = [
    'http://example.com/test1',
    'http://example.com/test2',
    'http://example.com/test3',
];

function processQueue() {
    if(tests.length > 0) {
        var test = tests.shift();
        $.get(test, function(response) {
            $('#some_ul').append('<li>' + response + '</li>');
            processQueue();
        });    
    }
}

processQueue();