I have just published on this topic but in the context of metabolomics, asthma, and vitamin D: http://ajcn.nutrition.org/content/early/2017/08/23/ajcn.117.158220.abstract

Determining the optimum number of k centers for a dataset is one of the most challenging questions. In my work, I went by 3 different metrics in order to help choose the ideal number of centers, namely:

• Gap statistic (Tibshirani, 2001) (bootstrapped 250x)
• Silhouette coefficient (Rousseeuw, 1987)
• Elbow method

To scale or not to scale? - that is the question

This is key. It depends on whichever method you are aiming to use. The Gap Statistic calculation, for example, involves a log and then single value decomposition transformation on your data, so, it needs to accept unlogged data. I have parallelised the R implementation of the Gap Statistic, called clusGap, here: Parallel processing in R

k-means or partitioning around medoids (PAM)?

Don't limit yourself to using k-means. PAM is a more flexible algorrithm and I argue is better for complex data. I used PAM in my published work above. When you find an ideal k and then run PAM with this value, it will return medoids:

Re-cluster using identified cluster centers

Yes, you can indeed later cluster your samples based on the identified number of cluster centers and the sample-to-cluster membership:

Hi Less,
You touch on some really important questions, the answers to which will depend on what works best for your dataset so I highly recommend trying things multiple ways and seeing if they line up with your expectations of the data. The method for estimating K can also give varying results so again try several ways. I recently wrote a tutorial for estimating K for RNAseq data which you could find useful!

For these questions I'm assuming you're referring to expression data.

1) Is it mandatory to scale the dataset before? If yes, why?

I personally think this is important if you want to extract clusters based on gene profiles and not gene expression values. Without scaling your highest expressed genes will cluster as one group, your lowest as another etc. Scaling allows genes of similar profile to cluster together regardless of their absolute expression level.

2) Is it recommendable to remove outliers before? If yes, what's the best method to evaluate which outlier to exclude?

Outliers removal should be very carefully considered and only for good reason. You can always run it both ways and compare. Outlier samples can be identified by hierarchical clustering of the samples, outlier genes is another story. People sometimes filter lowly expressed genes as noise. You can also filter the genes a posteriori using their correlation to the cluster mean.

3) I have seen that the estimated K could be used also for hierarchical clustering.

It is indeed possible to extract K-clusters from a tree. This can be used as a way to cross validate your clusters.

In k-means, the number of clusters k is a parameter given to the algorithm, i.e. k-means gives you the number of clusters you ask for. There is no universal way of determining k. The method mentioned at the bottom of the page you linked to uses the gap statistics to try and find a good value of k but there are other methods. You can use this value of k for other clustering algorithms that need the number of clusters as input, in the case of hierarchical clustering, this can be used to cut the tree into this number of clusters.
Scaling and outlier removal are data pre-processing steps. If and how they should be applied depends on the data and the question at hand.