As you have noticed, the Jaguar bridge has many moving parts. The height-adjustment screws for the individual saddles loosen and rattle, and so do the overall-height adjustment screws at the feet of the bridge assembly. Also the saddle sides rattle against each other. This is easily fixed by "affixing" these parts together with something. Fran, here, likes to use vaseline, or some such grease, which simply dampens the vibrations between loose parts, to stop the rattle. This is a good "purist" solution, since it doesn't really add anything to the mass or rigidity of the bridge (which would affect tone), and it retains it's full adjustability. When I used a stock bridge on my Jaguar, I used a hobbyists' water-soluable white glue ("Kristal Klear") to hold everything together, with good results. I've since switched to a Mustang-type bridge, as I like the increased sustain and "twang" it gives the guitar because of it's larger-mass saddles, with fewer moving parts. I still glue everything together with the Kristal Klear, though.
I've gotta say something here about Jaguar/Jazzmaster bridge placement: Having to constantly re-set your bridge in the perfect "upright" position after string changes, adjustments, and just hard playing, gives a certain imprecision to one's intonation which may not be acceptable in your style of playing. A sure-fire solution to the intonation problem when using the stock, or the Mustang bridges is to intonate the bridge saddles with the bridge pushed all the way back. This gives you a fixed point to be in tune from. It also means that your vibrato will now best work mostly for downward bending (although about a half-tone of upward bend is still possible without pushing the bridge forward and out of tune on return). I like to wrap about one wind of electrical tape around the bridge posts at their point of contact with the back of the bridge sockets to stop rattling there, too.
Thicker guage strings have more tension, and do a better job of staying down, and holding down the stock bridge and all it's parts- hence, less rattle/buzz. If you like "9's" or "10's" for strings, you can increase their length/tension using a buzz-stop. That's what they're mainly for.
I hope that this isn't too dense or too technical for you- Enjoy your new Jaguar!
