Dark brown sugar is more acidic than light, which may make your cookies rise higher/cakier. It is also a bit more hygroscopic from the higher molasses content, which can make cookies a little more chewy and a little less crisp. See "cookie fact #8" in this Food Lab test of cookie components; the comparison is white v. brown, but light v. dark will be similar but possibly less dramatic: http://sweets.seriouseats.com/2013/12/the-food-lab-the-best-chocolate-chip-cookies.html
How much these differences appear will of course depend on the type of cookies.
I have often bought dark brown sugar instead of light brown on accident (normally from Trader Joe's) and I use it the same, the cookies just have more of a caramel flavor which isn't a bad thing in my mind!
Yes you can- ordinary brown sugar (as opposed to raw sugar and such) is just white with molasses added. The molasses also adds some acidity, which will have an effect if there is baking soda in the recipe.
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How much these differences appear will of course depend on the type of cookies.