Interesting questions.
For your first case (uninitialized local variable), yes, the processor will "spend time" making room for your variable, but it's not very much time at all. It doesn't have to 'find' a spot...it knows exactly where to put it (on the stack, if that helps at all).
If you define and initialize the variable it does the same thing, but requires an extra step to set the variable to its initial value. A little extra time compared to the previous case, but still not very much time at all.