Can the theory of natural selection distinguish between two correlated phenotypic traits?
Are biologists aware of the causation vs correlation problem or are they noobs in statistics?


Evolutionary biologists tend to be pretty good statisticians. For historical consideration, Pearson is the father of the concept of correlation and was a biostatistician and Fisher is (one of) the father of evolutionary genetics and is also a famous statistician (invented the Fisher's exact test, t.test and Anova). Many of the modern developments in modern statistics were brought about by geneticists (incl. developments in HMM, ABC, FDR, etc...).

So, yes we are aware of the causation vs correlation problem.


  In the theory of natural selection, can the two situations be disentangled especially when one is talking about evolutionary accounts that are on the order of hundreds of millions of years[?]


Just like in any other causation vs correlation problem, no it is not easy to disentangle the two. If the specific hypothesis put under testing is subject to experimental manipulation, then yes, it is often feasible to disentangle the two. Otherwise, in a purely observational study, one can logically reason and argue for one rather than another causality relationship but in essence, you cannot know in a purely observational study whether a correlation is representing a direct causation.

The follow-up question might then be


  Is experimental manipulation feasible in evolutionary biology?


The answer is yes. Not for all questions though. 

First there are evolution experiments. An evolution experiment is just like what is sounds like. You put a population in a controlled environment and let it evolve. Repeat the process many time to have a decent sample size. There are also many question of interest to evolutionary biology that can be studied through experimental manipulation (and not only observatory studies) that do not require an evolution experiment.

You seem to be particularly interested in the reconstruction of past evolutionary history of a lineage (evolution biology is much more than a 'historic' reconstruction of life on earth), I would like to highlight that in purely observation studies and in experimental studies not involving evolution experiments, there are plenty of methods to reconstruct past events. Consider for example the loss of genetic diversity at linked variants (selective sweep) caused by positive selection. If two traits are correlated through time (which is very hard to figure out in absence of good fossil records) or correlated over several species (consider having a look at phylogenetic contrasts methods and more modern version of inference of selection over phylogenies), then if selection occurred in a single of the two traits, you should expect to see a differential loss of heterozygosity among the genetic markers explaining ariance for one trait and for the other.