How does your hand/stand mixer's speed effect the texture of your cookies? I've read many recipes asking to mix sugar and butter at medium speed for about 5 minutes. Can I use slower speed and mix it a little bit longer? Will it make any different in the end? 

One more thing, when mixing dry ingredients into wet ingredients, does it really make a difference if I use hand mixer instead of a spatula? In the videos, I've seen most of the bakers usually switching to spatula when they have to mix wet and dry ingredients together but some of them mix them in a stand mixer.
Speed would impact in so much as rapid speed helps incorporate air bubbles. These bubbles are important, since they capture and hold CO2 during the rising portion of baking (essentially I like to think about baking in 4 phases: mixing, heating and rising, setting, and cooling).

Usually you use a spatula vs a hand mixer when you A) are using an incredibly dense dough which could damage a mixer, or B) you are concerned about the formation of gluten. Gluten is essentially a good thing in breads, but a bad thing in cakes, and muffins.