Is there an analysis of World Champion games, with the help of engines, that shows statistics like percentage of best moves played, blunders made, creative moves which computers can only see at high depths, and etc? There might be some bias in this analysis since solid players, at least to me, will play objectively better moves since the positions that arise are simpler.
Yes, there are such studies. You can check this recent paper and references therein for the older ones. Various ideas have indeed been tried in order to quantify the "complexity" of position (e. g., in terms of complexity of search tree or dynamics of the evaluation function as the depth grows), but I think you better read the papers if you are interested in details.
Not that I know of. You could do this yourself for each world champion, taking a sample size of something like 20-30 games for each. That may give you fairly good statistics, since as the sample size goes above that there won't be any dramatic changes.
It's a good point to mention the bias, but in who's favour this works isn't clear. If world champions get in tougher positions, there tend to be fewer acceptable moves (meaning the number of choices to match with the engine's best choice is smaller). But in a solid position, there are often many acceptable moves (and you need to find the one the engine likes best).