From this output would it be possible to write an equation like with multiple linear regression?
So:
$$\text{Salary} =0.032\cdot \text{Gender} + 0.013\cdot \text{Ethnicity} + 0.097\cdot \text{Subject}$$ Or is it this not the right way to do so?
I have done extensive research and cannot find any examples similar to this.
Thanks in advance :)

It is OK as long as you remember that this model is with respect to a standardized data, which in your case is standardized salary $y$.
(i.e., $y$ is not salary per-se, but $y = (\text{salary}-\text{mean(salary)})/\text{SD(salary)}$).