What side effects should I expect by replacing cake flour with chickpea flour in all cookie recipes?
This article, from the Examiner, indicates:


  Garbanzo bean flour (a.k.a. gram flour, chana flour, besan, chickpea,
  or cici flour) should be a pantry staple. Unlike other bean flours,
  garbanzo bean flour does not have to be combined with other flours
  (although you can do it if you wish). Garbanzo flour is high in
  protein and gives a slightly “beany” flavor to baked goods.


However, they are not focused on food issues, and may not be the most reliable source.

According to Living Healthy Mom, you can use chickpea flour as a flour substitute, but they recommend no more than 75% replacement (garbanzo bean is another name for chickpea):


  Garbanzo bean flour- I know this gluten free flour doesn’t sound appetizing, but it is delicious, healthy and is a wonderful primary
  gluten free flour that you can use up to 75% in a recipe. And…………..NO,
  I know what you are thinking……… it doesn’t taste like beans! (I
  thought the same thing!)


Again, not the same as a university extension center, but at least a practicing individual who tries these kinds of things.

Given that cookies are made from a structure of starch rather than gluten, the lack of glutens from wheat flour should not be an issue.  However, chickpea flour also contains less starch 

One brand of garbanzo bean flour indicated that it had 18 grams of carbohydrates per 30 grams of product on its nutritional information; that same brand's pastry flour (a wheat flour) had 27 grams of per 34 g of product.  As you can see, chickpea flour has significantly less available carbohydrates (mostly starch, some sugars) to form the structure of the baked goods.  This will change the texture and structure development.

In general, gluten free baking experts recommend using a mix of flours to substitute for wheat flour, depending on how it is being used, in order to get the best possible outcome.  If your goal is not to be gluten free, only substituting for 50% of the recipe amount may provide good compromise on the outcome.

Bottom line, in a cookie, you would expect when substituting 100% chickpea flour for wheat flour:


Different texture, due to less starch and more protein.  My best guess is that it will be a little more fragile, due to less starch structure, but I have never tried this.
Different flavor, due to the fact that beans taste different than grains


Update after the question was edited to talk about the effect of leavening:

In a cookie, the structure of the final product is based on gelatinized starches.   With the reduced quantity of starch in the chickpea flour compared to wheat flour, there is simply less of a starch network present.   This may reduce the capacity of the cookie to retain the gas generated by the baking powder or baking soda, and thus a flatter, less airy product.