First, how it is possible to introduce an exogenous shock (i.e. random variable or exogenous process) with specific distribution assumptions?
For example, Assume that I have a linear function of the following form.
R1_t= m1_t + e1_t
R2_t=m2_t*e2_t
m=[m1,m2], is a vector of endogenous variable, and e=[e1,e2] the vector exogenous random variable. Further assume that, my distributional assumptions are, e1 is normally distributed with (mu1,sigma1^2) and e2 log-normally distributed with some mean and variance (mu2, sigma2^2) respectively. For simplicity suppose that these two shocks are independent, and e2 is independent from m2.
Dynare uses perturbation techniques that rely only on the first and second moments, i.e. essentially, you can only use normal distributions for exogenous shocks. But you can work with transformations of these exogenous shocks, which are then approximated to a particular order. For example,
will be log-normally distributed if e1 is normal. When going to 2nd order, part of this log-normality is preserved.
Regarding the second point, Dynare only accepts mean 0 shocks. Thus, you need to split your shocks into a deterministic mean component plus a random shock. But Dynare will not understand what you wrote down, because if m1_t is an endogenous variable, E_t(m_t+1) will not be m_t unless it follows a random walk.