1) How to find the best configuration of the sae: number of layers, number of neurons in each layer. Is it by trying different configuration and check for example the classification accuracy?

Cross validation (or any other validation technique).

2) Is fin-tuning always necessary?

Yes. In the preinitialization step, adding noise and trying to recover the original features in the output you initialize the weights of the neural network, but you never show the labels of the samples. Using fine-tuning after the preinitialization you train the network to solve a task (in your case to recognize numeric digits).

3) If I train the sae using say 60000 samples (like the MNIST example ), how often should I re-train the sae knowing that I am able to get additional data over time.

When your samples have new information but I don't know a rule to tell you that. You can use your neural net with your additional data, if your network classify correctly all these data probably you don't need to retrain it.

Thank you Roberto for the clarification.

About the second question I found in Stanford tutorial about sae and I am quoting" When should we use fine-tuning? It is typically used only if you have a large labeled training set; in this setting, fine-tuning can significantly improve the performance of your classifier. However, if you have a large unlabeled dataset (for unsupervised feature learning/pre-training) and only a relatively small labeled training set, then fine-tuning is significantly less likely to help."

So apparently, fine tuning does not always improve results and it depends on the data you have.