Because you say the recipes work fine when you hand mix them, I'll assume that your issue isn't aging ingredients or substitutions.
Honestly, I'd say that running it on the slowest speed is probably part of the problem. A higher speed for a shorter period of time is probably going to incorporate ingredients better because they get thrown about more.
With my Kitchen Aid, I generally start cookies by creaming the butter and sugar together on medium-medium high (6-8) for 2 minutes. Then adding the eggs and liquid flavorings (vanilla, almond extract, etc) and beating until combined.
In a separate bowl, I will have combined all of the "dry" ingredients (flour, leaveners, salt) by whisking them together to aerate the flour and mix them (I rarely sift flour).
When the wet ingredients are ready, I dump in the dry ingredients and start the mixer on "stir" to keep the flour from flying out of the mixer and, when mostly incorporated (10-15 seconds), I'll bump it up to medium low (4) to fully combine (20 seconds). If there are other ingredients like chips or nuts, those will get a quick stir in at the very end (5 seconds or less) with a last turn with a spatula to make sure everything is incorporated.
Using this method, I've never had an issue with tough cookies.
Cakes are more difficult to explain because there are vastly different methods but I'd encourage you to try a higher speed, shorter duration mix in of your dry ingredients.