In which statistical tests is it appropriate to use control variables that are not independent of the exposure variable? In my field, I frequently see researchers employ control variables which are not independent of the exposure variable.
For example, numerous studies have revealed that exposure to induced abortion is associated with increased substance use and more sexual partners.  But obstetricians, who may not be familiar with this research, have published a study examining the rate of miscarriage among women with a history of induced abortion controls in which they reduce the RR by controlling for the number of sexual partners (used as a proxy for risk of exposure to venereal diseases).  But if abortion contributes to both number of sexual partners (a possible psychological effect) and to miscarriage (a possible physical effect), the association between abortion and miscarriage may have multiple factors and pathways.  
I have even seen some studies that will use literally four or five "control variables" which have elsewhere been demonstrated to be significantly associated with the exposure variable.
In which statistical analyses, if any, is it appropriate to use as a control variable a variable that has itself been found to be significantly associated with both the exposure variable (being tested) and the outcome variable?
My preference, and frequent request, has been for researchers to show results in a table segregating the findings for each "control" group.  For example, in the analysis of miscarriage relative to exposure to abortion, I requested the results be segregated to show RR of miscarriage for women in bracketed groups based on number of sexual partners (SP), such as SP=1; SP=2-3; SP=4-5 and SP>5.  Such a breakdown would make it easier to see the relative effects of abortion and number of sexual partners on subsequent miscarriage rates.  (Unfortunately, the author refused to provide the requested segregation of results.)
So my question is what kind of analyses can properly account for control variables that are not independent of each other, and especially, are not independent of the main independent (exposure) variable being tested?  
In short, I'm trying to figure out if and when I should accept that an analyses is not confounding the results by using "independent control variables" that are not truly independent of each other.
 A: If I am understanding you correctly, then my answer is "pretty much any method, but the right approach depends on the goals of your research and the specific questions you are asking".
For example, suppose you are looking at the effect of diet on weight loss.  You should certainly control for demographic variables that are associated with weight loss, because those variables are going to help you understand what is going on.  E.g. people who are larger are probably going to lose more weight on any particular diet. 
When examining things that have multiple pathways and possible relationships, it is rarely correct to do just one analysis and which analyses are right depends on context.  When, as in your examples regarding abortion and substance use and so on, the study is impossible to randomize, this is even more true.
Deciding what question to ask is a substantive matter, not a statistical one.  And controlling for variables changes the question.  
A: In short, yes, it is very common and often desirable. 
The whole point of multiple regression analysis (for example) is to account for covariation between multiple independent variables and a dependent variable in order to determine the unique contribution of each to variation in the dependent variable. So, it is not at all uncommon for correlated independent variables to be used in the analysis. In fact, that's the point! If you know (or suspect) that a variable is correlated with your independent variable and your dependent variable, you better put it in you model, lest you bias your results. (Now, very highly correlated independent variables run into the problem of collinearity, which is a technical estimation problem and a substantive problem, but I'all leave that to the side.)
I'll give a different example. Let's say we know that class size and student test scores are correlated. We might conclude that we should reduce class sizes, because that should increase test scores. But family income is associated with both student performance and class size. What looks like the impact of class size may just be due to family income. Therefore, I want to add that as a control variable. If I do, then I get the association between class size and test scores that is not due to family income. However--this DOES NOT ensure claims of causality at all! I have just determined the association between test scores and class size net of family income. So if something is correlated with getting an abortion and having a miscarriage, then you definitely want to include it in your regression analysis. 
You ask when you should accept an analysis. There is certainly no rule of thumb. Is the analysis appropriate for the question? Is the analysis done well? Does it warrant the inferences that are drawn from it? Are causal claims drawn from correlational evidence? 
