We can only observe correlations
Let's just talk about statistics. You can see a correlation between two things only if there is variation for these two things. It therefore, make no sense to look at a single trait that has no variance and ask "is it genetically coded?". The only thing that makes sense is to understand what variables explain the observed variance for the trait of interest. Here, in your question, you are interested in behavioural traits. This answer stands for behavioural traits as it stands for any other phenotypic traits.
What variables could explain variance in a given trait?
To make it easy, let's consider only two variables, genetics and environment. In reality, one should consider other variables such as epigenetic for example. The total variance in phenotype that you observe is the result of the addition of the variance of this phenotype that is due to genetics variance ($V_G$) and the variance of this phenotype that is due to environmental variance ($V_E$) and their covariance ($COV_{GE}$). In other words:
$$V_P≈V_E+V_G+COV_{GE}$$
It is very common to express the phenotypic variance as being the sum of the genetic and environmental variances (and some others such as epigenetic variance). However, you are free to make this partitioning as you feel it useful for you.
Let's investigate those variables a bit further
Of course, $V_E$ and $V_G$ can be further decoupled into a sum of variances (and covariances). And you are free to do the decoupling exactly as you want to. For example $V_E$ could be defined as the sum of the phenotypic variance that is due to variance in temperature ($V_T$), the phenotypic variance that is due to variance in water availability ($V_W$) and the phenotypic variance that is due to variance in the culture of the parents ($V_P$) (and all the covariance). $V_E = V_T + V_W + V_P + ... $
Similarly, the genetic variance $V_G$ can be further decoupled. It is standard to decouple this variance in dominance vs additive effects (relating to the genetic architecture underlying the trait) but this won't be of great help to you. How about talking about variance in the genes affecting brain morphology $V_B$ and variance in the genes affecting hormones concentration $V_H$ and their covariance (epistasis and pleiotropy) $V_G = V_B + V_H + ...$.
Common schema
To my knowledge there is no common schema in how to partition those variances for the specific study of the mechanistic of behaviour. I don't really understand what is the partitioning that you're suggesting in your post but I am hoping that this answer can help you to think further and eventually come up with a satisfying schema.
