Is the random forests algorithm Turing-complete? As in, can any algorithm be represented by a given "tree" in the forest?
Random forests are useful only for inputs of a given size, so they are not really eligible for Turing-completeness. However, any function on inputs of size $n$ can be computed by a decision tree, so in that sense the model is complete (though in practice we limit the depth of the trees).
No, they're not Turing complete. The handle inputs of a fixed size. Also, they can't loop, so will always halt -- it follows that they don't have the ability to perform Turing-complete computation.