I have a Kona hardtail. I'm 5-8 and the bike is a 17" frame with a 100mm travel fork. It is very comfortable on long rides, easy to ride up hill. Downhill has been a real nightmare. I went over the bars on Wednesday night at a place that was really over my riding skill level. The rear brake wasn't slowing me so I used the front a little and the wheel caught.
When I look at my bike, the hand grips are the same elevation as my seat. Is this right? Should the hand grips be a little higher?
What things can I do to feel stable going down hill (not bomber fast)?
What things can I change to make the bike feel equal going up or down? Down is fun, but not when I have to worry about flipping over.
The biggest thing is just to increase your skill level.
More specifically, your riding position is really important. Lift your butt just off the saddle, so you have the freedom to move forward and back relative to the bike. Depending on what your dropping off, you may actually get all the way behind the saddle. I find that position impractical for a long descent, but as long as I'm hovering, not sitting, and well balanced, it doesn't matter - I have the freedom to put my weight wherever I need it, whenever it comes up.
EDIT: For general descending and singletrack riding, if you're hovering you should also make your "platform." Put the pedals at 9 o'clock and 3 o'clock. 6 o'clock and 12 o'clock is good for cornering, but it tends to bias the weight of the bike to one side, so it's not great if you're just trying to stay balanced.
I like to get low in the torso, so I have a fair amount of bend in the elbows. Having a lot of bend is another thing that gives me the freedom to get behind the saddle, or initiate a manual.
If your cockpit is sized wrong, getting in a good position is a lot harder. You might feel cramped, or you might not be able to get back far enough, depending on whether your cockpit is too long or too short. Having the bars level with the saddle is a pretty standard XC position, among riders with average male proportions and who aren't masochists. Shortening your reach or raising your handlebars will tend to put you in a more upright position in the saddle and facilitate being further back when you want to get behind it, but may mess up your handling on a climb. Lowering your handlebars or lengthening your reach may improve climbing slightly, although it sounds like you're pretty happy with that already, but it'll make it a lot harder to get the front wheel up.
Too soft a fork can also cause the bike to nosedive under braking or in a compression.
A really useful skill for descending is a Manual. They look sort of like wheelies, but it's done with a weight shift, not torque on the pedals. People who are good at them can sometimes balance on their rear wheels and do all sorts of crazy stuff, but all you really need to be able to do to have it make mountain biking more fun is get the front wheel up a little bit, so you land a drop in a more balanced position. IMHO, the best way to land a drop on a hardtail is on the rear wheel, with the front wheel landing almost immediately afterwards. Obviously you need to be off the saddle so you can soak up some of the impact with your legs.
It's also useful to practice a wheelie because, in my experience, they're easier to do at slower speeds but have a similar feel to a manual. You can work on wheelies and manuals someplace flat, like a park or the sidewalk in front of your house. I think different people find they prefer different skills; bear in mind that for a wheelie, you need to be in the "right" gear, so they're not always that useful for descending. There are good videos of both on YouTube. Everyone's approach is different, so if you don't get anything out of one video, watch another. (You do NOT need the trials or freeride bikes the guys demonstrating them typically have. Lowering your saddle can help to learn, though.) It's useful to practice wheelies going up curbs and and manuals dropping off them. If you can manual off a curb and land nicely, you're most of the way to being able to manual off something bigger.
Aside from making sure your fork spring is stiff enough, I wouldn't worry about buying a new stem until you've spent some time practicing. If your stem's not already flipped up and at the top of the spacer stack, though, you might do that, since it doesn't cost anything and is easy to reverse.