I don't know of an empirical analysis either. Some years ago I attempted to do one on my own but ended up frustrated. The 54% advantage mentioned in the answers above is just an average from all recorded games and most probably conceals important correlations. My hypothesis was that white's ELO-advantage is low for weak players and high for masters. I have the feeling there is a peak somewhere below the elite level. Who knows? But even with a database of ~10 million games the results remained inconclusive.
You have to group players according to their strength and then look at how they perform against other groups. For example: What performance do players from the group (2500, 2510] achieve with white against the group (2510, 2520]? The problem is, that if you make these intervals too big, the hypothetical rating advantage is lost in the noise. If you choose them too small you end up with too little games.
Your second formulation of the question might have a definitive answer although it probably depends on the ELO of W. But I doubt this formulation is equivalent to your first one. A player with a rating advantage of 100 points might profit more from the white advantage than a player with a mere 50 point advantage, while a player with a 500 point advantage will win almost every game independent of the colour they are playing.
Unfortunately you have to make numerous theoretical decisions before you can hand down your question to statistics.