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Let's say that I have a parameter vector of estimates, either estimated with the maximum likelihood command in Dynare or some other way. Now I want to make forecasts based upon that parameter vector. I also want uncertainty bounds around the forecast

Let's say that I have created posterior distributions of my own in regard to the parameters that I can use.
I want the uncertainty bounds to be based on.

1. Uncertainty based on the parameter uncertainty
2. The state variable uncertainty (Uncertainty from the Kalman filter calculations).
3. Uncertainty due to shocks
4. Perhaps uncertainty due to measurement errors.

There is a forecast command in Dynare, so the question is how much of the above (points 1-4) can I conduct by using built in Dynare functions? And what functions of Dynare should I use, more specifically?

Bayesian forecast after estimation will do all that. Classical forecasts will only take 3 into account. You could use Dynare routines to implement 1 to 4 with your own posterior estimates, but then programming the stuff yourself might be easier.

Measurement errors are generally not yet taken into account, see https://github.com/DynareTeam/dynare/issues/853.