What assumptions are necessary to establish that a variable is confounding? Suppose that an article claims that there is a cause-effect relationship between a certain explanatory variable A and a response variable B.
If I were to test whether a confounding variable C exists in the study, what are the statements that I would have to prove in order to establish C as a confounding variable?
I know that I must show that the following statements are true:


*

*Variable C must have an association with variable A.

*At the same time, variable C must have an association with variable B.

*Also, variable C must not be an intermediate in the cause-effect relationship 
(i.e. it cannot be A $\rightarrow$ C $\rightarrow$ B).
Are there any more statements I must prove in order to establish variable C as a confounding variable?
Can someone provide an example of a well-established confounding variable?
Also, just a side question, is it possible to have a confounding variable in an experiment? Or, do confounding variables only appear in observational studies?
 A: Those statements are not enough because they are too weak. Association is not enough to determine confounding; you need causation, which has within it a temporal assumption.


*

*C is a cause of A. 

*C is a cause of B beyond its effect on A.


An example in my area of research (higher education) is the following: for the causal effect of a highly selective academic enrichment program on college GPA, high school achievement is a confounder. High school achievement causes selection into the program. High school achievement causes variability in college GPA, beyond its effect through the program. Without adjusting for high school achievement (e.g., by regressing college GPA on program participation and high school GPA), a possible observed difference in college GPA between program participants and non-participants could be attributable not just to program participation but also to high school achievement, thereby underestimating the unique effect of participation.
A: 
I know that I must show that the following statements are true:
  
  
*
  
*Variable C must have an association with variable A.
  
*At the same time, variable C must have an association with variable B.
  
*Also, variable C must not be an intermediate in the cause-effect relationship     (i.e. it cannot be A $\rightarrow$ C $\rightarrow$
  B).

This is not correct, you should check some of the things here, here, and here. For example, your statement would fail to recognize a collider, where all these statements would be true, yet $C$ would not be a confounder. So you might want to adjust your claim by saying, for instance, that $C$ is a common cause of $A$ and $B$ and its confounding effect was not blocked by the study's design.  

Can someone provide an example of a well-established confounding
  variable?

Yes, just think of ice-cream sales and diseases that are more common during warm seasons. These things will be correlated due to the common cause (warm season). 

Also, just a side question, is it possible to have a confounding
  variable in an experiment? Or, do confounding variables only appear in
  observational studies?

Yes, it's possible to have confounding in experiments. The difference between observational studies and experiments is of degree. In an experiment you hopefully (physically/by design) control/block some of the confounding factors and randomize to account for the ones you can't control. But you can only do this up to a certain point. After randomization patients might not follow the protocol, you might have patients that drop the study, etc. Several things might happen. You might also want to check this answer here.
