Can I make a cake with finely ground coffee instead of cocoa powder?
Coffee cake is good, but it's better to start from a recipe for coffee cake than try to adapt a chocolate cake.  Cocoa powder brings an appreciable amount of fat and bulk to the recipe. 

You need to be very careful about terminology.  Coffee powder in many places refers to powdered instant (soluble) coffee, which isn't the best coffee drink but is very suiatable as a flvouring.  It should be dissolved in liquid before adding tot he cake mix (following the recipe).  I have seen coffee powder used to refer to finely ground coffee beans.  It's not unknown to add these to food, but it's not common.  We discussed this recently.