Why is it so much easier to fall when walking down a slope? When I walk uphill on an icey road, I almost never slip/fall. But if I walk downhill or even on a relatively flat road, I slip/fall frequently.
What is the physics reason behind this, and are there any tips to reduce the chance of falling (other than shoe modifications and "being careful")?
 A: I think it is better to say that concept of torque is working here. 
When you walk on floor, there is no force which can provide a torque to your body. But when you walk on a slope,  a component of gravitational force provide you a torque which results in falling.
A: There's a whole slew of reasons, as seems to be true for all biophysics issues.  However, they generally all center around stability.  When everything goes perfectly according to plan, you don't fall.  Its when your path is perturbed slightly and you have to make a correction that you get in trouble.
The major reason we fall when going down hill is that we actually fall as part of our step.  When we're walking up hill, it's easy to move one leg forward, bend it while it is unloaded in the air to reach a higher point on the hill, put it down, shift our CG on a path across the leg, and when it gets almost over the leg, we do a leg extension to lift ourself up.  Going down hill, however, is harder.  To reverse the process, we would have to bend our back leg, while under load, to move our entire body down such that the front foot can touch the ground below us.  This is an eccentric motion (motion is in the opposite direction of the force), which is harder for us than normal motion.  Also, we can't overshoot when we shift our CG, because we would fall forward.
As a result, most of us choose to fall down a hill.  We lift our back leg up, swing it in front of us, and then pivot around the back foot, falling forward until the front foot slams into the ground letting us arrest the fall.  We have a little more finesse than this, but it's the general shape of the motion.  This is the motion which makes going down hill so hard on your knees, because they take a blow every time you step.
Also, if we fall into our step, we have to worry about dynamic friction.  As we land, we have to arrest any forward motion.  If the ground gives way beneath us, we begin sliding with nothing but dynamic friction to slow us down, and dynamic friction typically has a lower coefficient than static friction.  When going uphill, without the fall, we can take time to plant our foot carefully and remain in the static friction region with its higher coefficients.
Some people are trained not to fall in their step this way. Martial artists, for instance, are trained to never fall into a step because once they commit to a step they are predictable until they land, and predictability can be the death of you.  However, even in this situation, going up hill is naturally more stable.  When going up hill, the hard part of the step stability wise is the start of the step.  Once you start stepping, you're moving up hill of your solidly planted foot.  If anything goes wrong, you can always fall back to your back foot, all the way up until the point where you start lifting it.  When you're moving down hill, if anything goes wrong once you start moving at all, your CG is now on the downhill side of your one stable foot, and pulling away from you.  You must get that front foot in place to stop this downhill movement.
One final detail is the way our foot is designed.  We are most stable if our weight is roughly centered on our feet.  When walking uphill, this is easy.  If we don't have the muscle strength and flexibility to step with the center of our foot, we can just let our toe rest on the hill and then start to shift our weight to that foot, which will naturally extend the tendons until the foot is firmly in place.  Going downhill, this process is harder.  If we don't have the skill to pronate our foot sufficiently to land flat and centered, we have to land on our heel and then rotate the foot down around that.  A quick look at our foot as a lever should show why this is troublesome.  Instead of properly shifting our weight to be centered on our foot, we tend to go downhill by landing on our heels.  Doing so prevents us from leveraging our calves properly.  They're our most amazing shock absorbers ever, strapped to the massive Achilles tendon.  Without the power of that shock absorber, we have fewer tools at our disposal for arresting any surprise movements we may endure during the process of landing.
Interestingly, the easiest way to go down something is to walk backwards.  It's much slower, and its a bit daunting to walk without being able to see where you're going, but it puts the body into a position where it is more inclined to step in the reverse of the uphill step.  You're more likely to step in the way a martial artist would want to step, and less likely to fall.  When you do fall, most people will fall to their knees, rather than tumbling feet over head.  You'll naturally try to use your calves properly, you'll naturally do the eccentric reverse-leg-press with the uphill leg to lower the downhill leg into a solid position.  And, if anything goes wrong, your hands are much closer to the surface to help resolve the issue.
A: When walking down an incline there is greater acceleration due to gravity. In order to maintain a walking speed, a braking force is needed to counteract the acceleration, and this braking force is given by the contact force of the initial footfall of each step as it's in front of your center of mass creating a force vector pointing away from your direction of acceleration. When going down a slope it's much more difficult for your initial footfall to be in front of your center of mass due to the limited reach of your legs as well as it being easier for your center of mass to pass in front of your supporting legs creating unbalance and downward acceleration. In addition your legs will more often be behind your center of mass than in front creating additional torque on your body that also has to be countered by your braking force.
There are several ways to counter each of these variables that cause you to lose balance.

*

*Taking shorter steps will broaden the range of your braking force vector (relative to your center of mass) and will help prevent overextending your legs to better prevent your center of mass from passing too far in front of your supporting legs. Depending on your speed this may not be viable since shorter steps will require an increased frequency of steps.

*Lowering your center of mass will also help as then your legs can reach the ground sooner than from a higher position and thus create a braking force sooner and stronger with a broader range for your braking force vector and increase the distance your center of mass has to travel before being in front of your supporting legs.

*Debatable, but walking backwards may help as some may find it easier to lower your body closer to the ground with one leg supporting your center of mass and have one leg extending backwards to provide the braking force than facing forward and having one leg extended forwards. The downside may be that if the torque on your body is great enough to cause you to rotate backwards it may more difficult to catch yourself and recover than if you're rotating forwards. This likely varies per person and the slope/walking situation at hand.

A: When you fall forward, your body instinctively catches you by moving the foot in front to push you back up. 
The foot has further to move before reaching ground on a downwards slope. 
A: I'd have to draw diagrams and such to explain to actual physics of the situation but long story short walking downhill puts more stress on ankle strength and upper body balance. Walking uphill allows you to use your quads more and since they are bigger muscles you have more control.
