VARs are the multivariate version of univariate time series regressions.
Other assumptions of OLS still assumed to hold. A VAR can be written without vector notation as a system of k equations. A VAR will be stationary so long as the modulus of each of the eigenvalues of A is less than 1. When you calculate the response of the system to a shock, you are calculating the impulse response function . The shock could also have been from an exogenous variable added to the VAR system rather than from the realization of the error term.