Regarding your first question regarding fireworks mode vs sports mode for shooting fireworks:
The key to a nice photograph of fireworks is capturing the entire blast, so it looks like a big, bright chrysanthemum flower in the sky. A fireworks blast is nothing more then several balls of bright light radiating from the center of an explosion in the sky. As those balls radiate from the center of the blast, our eyes perceive them as trails of light. In order for the camera to capture those flying balls as long trails of light it needs to use a long exposure, aka a slow shutter speed. In other words, the camera's "eye" needs to stay open for at least a whole second or more. The problem with such slow shutter speeds is that the camera needs to be perfectly still, or else the entire picture will be blurry. When I say perfectly still, I mean perfectly still, as in using a tripod. It's impossible to hand-hold a camera perfectly still. A hair's breath of motion, imperceivable to us, will cause blurry pictures at slow shutter speeds. The fireworks setting of your camera uses a slow shutter speed, so you need to stablize the camera by using a tripod or by using the camera's auto timer feature and setting it on the top of a trash can or something stable.
The sports setting of the camera uses a fast shutter speed. That mode is intended for freezing fast action in bright daylight. Because of the fast shutter speed, hand-holding the camera doesn't result in blurry images. However, when the sports setting is used on fireworks, it freezes the motion of the fireworks blast, resulting in an image of bright stationary dots in the sky, rather than the long beautiful trails of light that you'd prefer. So, when hand-holding the camera you'll get sharper pictures in sports mode, but your pictures of fireworks will suck. The better option is to use fireworks mode and a tripod (or some other way of stabilizing the camera).
Regarding your second issue:
Flash is generally only effective on subjects within 10 feet of the camera. When you take a flash photograph, the camera may think that the subject is within 10 feet and that it wrongly assumes that the subject will be properly exposed by the amount of light the flash it emits. That's why the images come out dark. When you take a picture without flash the camera knows that it can't rely on adding light to the scene, so it resors to using a slower shutter speed or a larger aperture (it opens its "eye" wider) to capture more light. That's why some pictures with flash appear to be darker than images taken without flash. However, indoor images without flash can be blurry because of the slow shutter speed.
Additionally, if you're using flash and the subject is within 10 feet of the camera. it's possible that the flash correctly exposes the person, but everything else in the room is very dark...it looks like the person is standing in a cave. Again, that's because the flash is limited in its effect on the rest of the room which is farther away than the person.
