What you say about noise is sort of true but not exactly. The pdf of noise for two added signals that have independent noise is the convolution of the two pdfs. For a Gaussian distribution this means that the variance of the sum is the sum of the two variances. For other distributions, you'd need to convolve their pdf's to see the distribution, but it will be wider. (By the way, this is the basis of "The Central Limit Theorem" that states that if you add enough noise sources they will approach a Gaussian no matter what their individual pdf's look like.) If the signals are not independent, like if you got your background signal from blurring your main image and you subtract it from your main image, then the noise actually is less. You can prove this mathematically and is the basis for unsharp masking.
For image division, the noise amplification depends on the amount you divide by. In the middle of the image, you won't introduce much noise but at the edges where you have more shading and vignetting, you will get more noise amplification because you have to multiply the image by a bigger factor there to flatten it. So the overall noise amplification very much depends on the shape of the "flat" background image. I can use a crummy $160 lens and get a lot of shading, or a so-so $700 Canon lens and get a fair amount of shading, or an excellent $1700 Schneider lens and get almost no shading (true examples from my experience). So your best bet is to just measure and see since the math to predict it depends on the amount of shading and that varies pixel by pixel.
