non stochastic regressors and causation Randomized controlled experiment is base case for causality (also) in regression. 
However currently I’m analyzing  the role of causality in linear regression as shown in many econometrics  textbook. For example in Brooks 2014 – introductory econometrics for finance 3rd edition pag 76-83 the fixed (non-stochastic) regressors are the base scenario and causal interpretation is explicitly offered. In this book the causal interpretation of regression coefficients seems  the basic scenario too. 
The example (pag 83) is about the CAPM and in this setting experiment, also ideal, and/or potential outcome language, at least in my experience, don’t play any role. 
I have several doubt but my questions are primarily three:


*

*Fixed non stochastic regressors assumption produce (by costruction) independence between errors and regressors. Then, hypothesis of stochastic independence between errors and regressors is equal to hypothesis of fixed non stochastic regressors ?

*If no (as I think see also here  regression and causation) the hypothesis of fixed non stochastic regressors is equal to known the “true model / true data generating process” ?

*Known this “true model” is, at least in certain sense, equal to construct an (idealized) randomized controlled experiment?
 A: 
Fixed non stochastic regressors assumption produce (by costruction)
  independence between errors and regressors.

This is not true, fixed regressors assumptions just mean that the regressors are not "random". You can still have bias, as showed in here. 
If you read a couple pages further in the very book you mention, Brooks (2014), he has a session on "omission of important variables", in which he mentions omitted variable bias (although I would not recommend this book if you want to learn causality, if you want to learn causality--and not just linear regression--start here).

The hypothesis of fixed non stochastic regressors is equal to known
  the “true model / true data generating process” ?

No, this hypothesis has more to do with the mathematics of the probabilistic manipulations. If regressors are fixed, you will treat them as "parameters", not as random variables. But this has no bearing on causal effects.

Known this “true model” is, at least in certain sense, equal to
  construct an (idealized) randomized controlled experiment?

Most authors do not really define what "fixed regressor" means except that it is treated as a fixed known quantity. Even when some authors do make the analogy to "fixed by experiment" (such as Seber, in a parenthetical remark), they do not offer any mathematics of causality. 
In short, to make causal conclusions you need to make explicit causal assumptions, as in a causal model.
