Can I force Flash games to play in fullscreen mode?

If you play in a browser, you culd use a bookmarklet or user script to full screen the game element. Try pasting something like this in your address bar:

javascript: addEventListener('click', function(ev) {var el = ev.target, rfs = el.requestFullScreen || el.webkitRequestFullScreen || el.mozRequestFullScreen; rfs.call(el);});

This should fullscreen a clicked element.
In some browsers (cough Chrome cough), you'll need to type the "javascript:" part yourself. Put it in a bookmark to circumvent that.

You can't. Normally, a Flash object stretches to fill all space given to it in the <object> or <embed> container, stand alone Flash player window, or screen. If it doesn't, there's nothing you can do without access to the original .fla file.