Is it possible to set PHP session variables using Javascript?
The session is stored server-side so you cannot add values to it from JavaScript. All that you get client-side is the session cookie which contains an id. One possibility would be to send an AJAX request to a server-side script which would set the session variable. Example with jQuery's .post() method:
$.post('/setsessionvariable.php', { name: 'value' });
You should, of course, be cautious about exposing such script.