Hi darkness.
Quick Exposure Basics
The problem you're experiencing is simple to understand. Firstly, as background info in case you're new to the concept of exposure, generally in any auto-exposure mode, the camera will attempt to calculate the correct exposure required for a scene and then suggest the necessary settings (aperture/shutter speed, possibly also ISO) to achieve the required exposure. However, do note that aperture and shutter speed work on a reciprocal basis, i.e. the higher the aperture, the lower the shutter speed. To be more precise, if you open up the aperture by one stop (i.e. divide the f-number by a factor of 1.414), you must "stop down" (make faster) the shutter speed by one stop (i.e. halve the shutter speed duration), or vice versa, given a constant ISO setting.
Your problem
However, all cameras and lenses are limited in the range of shutter speeds and aperture settings that they can use. Let's say, you have a scene that requires an exposure of f/2.8 (aperture) and 1/500s (shutter speed) in order to obtain "correct" exposure at ISO 100. However, if you use shutter priority mode and set the shutter speed to 1/1000s, you are doubling the shutter speed (stopping down by one stop), and therefore you need to correspondingly open up the aperture by one stop, i.e. bring it to f/2.0 in this case. That is to say, f/2.8+1/500s is equivalent to f/2.0+1/1000s (at the same ISO). However, if your lens cannot use an aperture size of f/2.0, then there is no way to obtain correct exposure at a fixed shutter speed of 1/1000s. Therefore, the best the camera can do is f/2.8 and 1/1000s, which is one stop away from the correct value of f/2.0 and 1/1000s and hence the result is underexposed since only half the needed amount of light reaches the film/sensor. This is probably what happened in your first case, with the added factor of the EC setting you dialled in (which I will try to explain later).
The second case is the same problem, but the other way around. By setting a slow shutter speed of 1/10s, you will require a smaller aperture size (bigger f-number) to achieve the same exposure. However, your camera is limited to f/8.0 at most, so it was not able to bring the aperture down to a sufficiently small size and therefore too much light reached the film/sensor, hence overexposure.
Now for EC. Exposure compensation is nothing more than a manual setting to override the camera's metering by a specified amount. Let's say the camera meters the scene and determines that an exposure of f/5.6 and 1/500s will achieve correct exposure. If you have an EC setting of +1.0EV, you are telling the camera to take its metered value and open up further by another stop, hence the camera might give you f/4.0+1/500s or f/5.6+1/250s instead. Likewise, with an EC setting of -2.0EV, the camera will meter the scene as per normal, but then close down its metered values by another 2 stops, hence you will get something like f/11+1/500s or f/5.6+1/2000s (if 1/2000s is available on your camera).
So what can you do?
While I don't have an S30, I believe most cameras will have some way to indicate when they are unable to achieve correct exposure (my Olympus camera will flash its exposure values in red). As such, for your first case, you can lower your shutter speed until the camera indicates that the auto-exposure is ok, i.e. the corresponding required aperture size fits within the range allowed by the lens. Alternatively, you can raise your ISO setting so that less light is needed to achieve correct exposure, but do note that image quality will tend to be compromised somewhat when you do this.
The second case is more tricky because you want a slow shutter speed, but it is not possible to use a slow shutter speed without a very small aperture in bright daylight conditions. What most photographers would normally do is to attach a neutral density (ND) filter to reduce the amount of light entering the lens, so that a larger aperture size can be used in conjunction with a slow shutter speed. A polarising filter will also tend to have the same effect, although that is not its primary purpose. Other than that, there really isn't any way to achieve your desired effect, as far as I know.
Apologies for the slightly verbose description, but I hope this helped in some way.
