If I load a PHP page with Jquery .load(file.php), can the included file use the php variables that were defined on the page that called the load()?
No, you have to pass the variables you want to use to your file.php:
$('#yourdiv').load('file.php?var1=xyz&var2=xyz&var3=xyz');
And then you can GET those in your file.php:
$var1 = $_GET['var1'];
$var2 = $_GET['var2'];
$var3 = $_GET['var3'];
If there are a lot of variables then use the POST method:
$('#yourdiv').load('file.php', {var1:x, var2:y, var3:z})
And then get the variables in file.php:
$var1 = $_POST['var1'];
$var2 = $_POST['var2'];
$var3 = $_POST['var3'];