Can Dynare++ calculate likelihoods for models solved above third order? If the answer is yes, I am assuming its a solution obtained via pruning?
There is no closed form solution for likelihoods at order>1. Thus, you need to use e.g. the particle filter. It is not implemented in Dynare++. As the particle filter is simulation-based, people typically also use pruning.
The particle filter is implemented in Dynare up to second order?
Yes, at third order, you would need to do e.g. the simulated method of moments.