1) Yes, absolutely. For example, if you have too less neurons in your hidden layer your model will be too simple and have high bias. Similarly, if you have too many neurons your model will overfit and have high variance. Adding more hidden layers allows you to model very complex problems like object recognition but there are a lot of tricks to make adding more hidden layers work; this is known as the field of deep learning.
2) In a single layered neural network its generally a rule of thumb to start with 2 times as many neurons as the number of inputs. You can determine the increment through binary search; i.e. run through a few different architectures and see how the accuracy changes..
3) No, definitely not - each hidden layer can contain as many neurons as you want it to contain. There is no way other can experimentation to determine their sizes; all of what you mention are hyperparameters which you must tune.
Im not sure if you are looking for a simple answer, but maybe you will be interested in a new neural network regularization technique called dropout. Dropout basically randomely "removes" some of the neurons during training forcing each of the neurons to be good feature detectors. It greatly prevents overfitting and you can go ahead and set the number of neurons to be high without worrying too much. Check this paper out for more info: http://www.cs.toronto.edu/~nitish/msc_thesis.pdf