Your headset's loose. It's easy to fix. You have to loosen both the bolts that clamp your stem to the steerer, and then tighten the bolt on the headset's top cap. Not too much though, just get it so that it doesn't rattle anymore, and then maybe go a tiny bit more. Then tighten the two steerer clamp bolts on the stem again, making sure the bars are straight of course. Those two bolts provide the strength that holds the whole thing together; the top cap just gets the stem into the right place.
Park Tools website will have a slightly more detailed writeup with pics if you need it.
The top cap should be what's putting downward pressure on the stem. Are the two steerer clamp bolts totally loose?
It might be that your steerer is a bit too long for the stack of spacers you have on there. Try adding another spacer to that stack... the spacer stack is what the stem uses to press down on the headset, and if that stack is shorter than the steerer tube, then the top cap can't press down on the headset.
You can definitely squeeze it all together by hand for a temporary fix (it's loose as hell and you might damage the bearings if you ride it around like that for long), but you should set up your headset so that the top cap works as it's a much easier and more precise way of adding pressure that's also independent of the process of aligning the bars with the front wheel.
If nothing else is working, your star nut - the thing inside the steerer that the top cap bolts to- may not be set deep enough inside of the steerer to allow the top cap to put pressure on the stack and stem.