It's not mandatary, but for some problems such as using image data, it provides as substantial speedup when training a classifier.

You could probably train MNIST on your Macbook Air but anything much more complicated for that you would want to use A GPU.

I believe that's a "no". I was able to set up a docker'ized Deep Style on my Macbook Pro, although it takes for bloody ever to do a single image. CUDA is, AFAIK, a substantial speed boost, but not a requirement.