Do digital cameras have an option for shutter speed?

If you know how regular cameras work, you also have an idea how the digital ones work,... sans, film. Instead of film, a light sensor (CMOS and CCD) are used in digital.

Most cameras give some control of speed, either using ISO (International Standards Organization), an acronym that in photography, is accepted to represent sensitivity to light based on international standardization or by controlling the amount of light striking the sensor over a given time (shutter) through a given opening (aperture).

Virtually all dSLR's and most P&S cameras give you one or the other.

Using distance as your determining factor for illumination of a subject will invariably affect the quality of the image only if by coming closer, you increase the overall coverage in the frame thereby increasing the interpretation of the light sensed.

With the inverse square law, light falling off from a flash is inversely proportional to the square of the distance from the source so an object (of the same size) twice as far away, receives only 1/4th the light energy in air.

Effectively speaking then, you are correct to make the judgment that if your coverage using a flash at a certain distance doesn't suit the exposure you need, you would just walk closer to or away from the subject to correct for proper exposure.

Knowing your flash's Guide number will allow you to make nearly instant calculations on correct exposure for your flash based on ISO, distance and f/stop.

It depends on the camera. Most of the digitals that are $100 or less you can not change the shutter speed, but most of them have options of how to shoot, based on what you are taking a picture of. I use a Canon Rebel that allows you to change the speeds, but that is a more expensive camera.

Nope, not with that budget. However, there are some point-and-shoots with scene modes that are called "night scene" or something similar that use slow shutter speeds. If you'd be willing to spend a bit more ($30-$50), you can consider looking at the Fujifilm FinePix S1800. It has aperture and shutter priority modes. Also, one of its key features is the 18x optical zoom. It costs around $180-$200.

Do you mean shutter speed control in Point & Shoots? Nowadays even most of the P&S cameras come with shutter speed control.

Back light (in the LCD) or IR have little to do with exposure.

most Canon powershots have a shutter speed option. Mine does (it's a powershot A720 IS). They range from low/mid $200 and up. I'm not sure about other brands, but generally with a lower end point and shoot you're not going to get that feature.

Yea they can but its usually the more expensive camera's that do that.

Depends what camera you have...