Also the post you referenced is for the case where the steady state does not need to be recomputed. Is there an additional step if I recompute the steady state? Or does Dynare handle it when I do stoch_simul(])?

If you are changing the variance of the shock, look at the .m file created from the .mod file. In addition to stoch_simul, I had to update the variance-covariance matrix of the shocks M_.Sigma_e.

You are right. The set_param_value command only refers to structural parameters. You cannot set standard deviations or measurement error defined in the shocks-block this way. You either have to set M_.Sigma_e directly, or calibrate the shock variances to 1 and use in the model

instead of

The structural parameter sigma_eps now contains the standard deviation and can be set using the set_param_value command. I have updated the referred post to reflect this information.