Shoot at the maximum resolution of the camera unless for some reason you're trying to save space on your memory card. Lowering this resolution will not and can not improve your images. It won't affect exposure or the camera's ability to resolve detail. The sensor always captures the full resolution, the setting only alters what gets recorded to your memory card in the final stage - your camera will do a size reduction.

If you intend to have some choice in what to do with the images at some point in the future, you want them with as high a resolution as possible. That will let you apply different reduction or filtering options on a computer to achieve the best results possible for the intended application. You can batch automate a reduction as a second set of images for instance if you want to give them to someone for email or other purposes where highest quality doesn't matter.

I took a look at a couple of the images posted. They look fine. As expected anyway. You ay benefit from some white balance tweaks as well as focus and reducing camera shake. You might also benefit from pushing the exposure on some of them. In the images I saw, the white envelopes in the foreground were blown out, the shot included elements at a very wide exposure which your camera just can't capture. You'll need to supplement the lighting of the darker parts of the image if you want to equalize the exposure over the whole frame.

Best noise results will be achieved when shooting at the camera's native sensitivity setting. So set the native ISO for best results in that regard. However, the best picture is a picture taken/captured. If you don't push the sensitivity in some instances, you might not be able to get an image at all. So you decide if you want a less than stellar image or no image at all in those situations.

You probably shouldn't expect to be able to handhold the camera for steady images at a shutter speed lower than 1/60th of a second. Even at that speed you'll probably still shake a few of them. It's inevitable with a small camera and the inability to properly support it like you would with a larger and heavier camera body. If you're using flash in a room dark enough to otherwise be severely underexposed (or not exposed at all) at the current shutter speed, then you shouldn't need to worry about shake. As long as the aperture opens enough to expose the scene with the flash, the shutter speed is going to be somewhat irrelevant (a longer shutter will only expose the ambient lit scene).

If you do want to replace your camera with something small, though not as small, and with better quality and controls, take a look at the Canon G11 or similar.