Hiya iamthelvy,
Yeah it should fit a 99 Fender then no problem. Tuning stability seems fine once you've set it up properly, which doesn't take long. There's a little screw on the back which you adjust to make the drop exactly the right length, and then off you go.
You're right too - it doesn't need screwing in. The screw collar of the machine head holds it tight and there are two little notches underneath a Fender machine head on back of headstock which this sits in to hold position.
It's a good bit of kit and I've had it for a couple of years. I just don't really use it much any more and trying to raise funds so this has to go!