See section 4.6 of the DYNARE reference manual on Auxiliary Variables for DYNARE implementation and the discussions on these forums here and here for examples.
My understanding is that anticipated shocks in DYNARE are by default modeled as in Schmitt-Grohe and Uribe (2012) and detailed in section 4 of their web appendix. If a shock is anticipated n periods in advance then DYNARE will generate n auxiliary variables to implement it. For example the paper above contains 7 driving processes which are subject to shocks anticipated 0, 4, and 8 periods in advance; as a result there are a total of 7*(0+4+8)=84 auxiliary state variables created in DYNARE if you enter the processes as x = x(-1) + ε0 + ε4(-4) + ε8(-8).
An alternative which may reduce the number of state variables (which is also documented in Section 4 of the technical appendix to SGU above) involves writing the states as a recursive system. This is my preferred method as it allows greater flexibility and often fewer states which speeds up estimation. But for stochastic simulations they are entirely equivalent.
All solution methods should be the same in DYNARE. The only difference is that with leads/lags in excess of 1 the program goes “behind the scenes” to replace what you wrote in the .mod file with the auxiliary state variables and their supporting system, and then perturbs this augmented system. Anticipated shocks aren’t actually any different than unanticipated ones, so I wouldn’t expect higher order approximations to present any special challenge.