The light meter is an incident meter, meaning how much light is falling on a subject, not how much light is reflected. Like how a compass tells you where north is, but does not tell you which way to go, a meter tells you how to set your camera to render a middle tone as a middle tone, but does not really tell you how to set your camera. You use the information the meter gives you to make an exposure determination. Sometimes this is what the meter reads out, sometimes its more, sometimes its less.
As for blown whites, if these are shiny surfaces, then the specular nature will lend them to "overexpose" and blow out. If it is a white shirt or dress, and it is copmletely blowing out, we may need to look at that further. How do the middle tones appear (i.e. non shiny surfaces such as clothing and dry skin)?
As to the back ground. If a black background is lit with additional light, it will turn gray. If it is lit with enough light, it can even turn white, so this is not surprising. The good news is it can be controlled.
The inverse square law, as applied in this case, states as you increase the distance from the light source, the intensity of the light will reduce by the square of the increase (in layman's term, if you double the distance, you reduce the light by 3/4).
So, if your subject is 3 feet from the back ground, and the light is three feet from your subject, the back ground is getting 25% of the amount of light your subject is getting. This is enough to make it gray. If you move the light to 1 foot from your subject (and reduce the output or your aperture to maintain the same exposure), the background will get 1/16 the amount of light, rendering it much darker.
If you were to keep the light 3 feet from the subject, but move the subject so they are 9 feet from the back ground (the output of the light nor the aperture will need to change as the flash to subject is constant), the back ground will get 1/16 the amount of light.
You can also use light placement, grids, barndoors, or flags to direct the light away from the background. If you want to ensure the BG will remain black, you can meter the subject and the BG and ensure the difference in metering is 3-4 stops difference.
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