Does whitening with PCA correspond to power spectrum flattening?
I believe the answer here is no. 

Remember what 'whitening' is really doing: It is ensuring that the power/variance that correspond to each of your basis vectors are the same. Which basis vectors? The ones you chose. 

In the DFT, you are choosing basis vectors comprised of complex exponential functions. In PCA, you are determining the basis vectors empirically. What is the relationship between complex exponential bases and empirical bases? You cannot say. 

Thus, when you say you whitened in the spectral domain, you are really saying "I made sure the power/variance associated with all my complex exponential basis vectors representing my data are the same". 

When you say you whitened in the PCA domain, you are really saying "I made sure the power/variance associated with of all my empirically derived basis vectors are the same". 

Are your empirically derived basis vectors complex exponentials? There is no guarantee. Thus, whitening in either domain is not equivalent to whitening in the other.


  If so, are the eigenvalues of the data covariance (which I understand
  are also referred to as a spectrum) directly related to the mean power
  spectrum of my signals ? How ?


If I understand your question correctly, then the answer to this is a simple yes, for the simple reason of energy conservation. All the eigenvalues together will give you the total energy/variance in your data. The average of this energy, will have to correspond to your mean power in any domain, including the spectral. 

Projecting your data onto either the DFT or PCA (empirical) bases dones not add/remove energy, but simply redistributes it. Thus, your average power has to always be the same.