No, it is not always possible, because it is not always possible to partition the (cyclic) sequence of edge-length into 3 portions of adjacent edges, for whose sums the triangle inequality holds; e.g. 1,1,3 doesn't allow the construction of a polygon.
Your condition of constructibility missed the cyclicity e.g. 1,3,1 meets your condition, but no triangle with that sidelengths exists.

If a polygon however exists, then there are several ways of making it unique, even in the convex case; you could ask either for minimal or maximal area or, you could demand its corners to be cocyclic, to name just some.

No, it is not always possible, because it is not always possible to partition the (cyclic) sequence of edge-length into 3 portions of adjacent edges, for whose sums the triangle inequality holds; e.g. 1,1,3 doesn't allow the construction of a polygon.
Your condition on existence suffices, as has been already demonstrated in the comment of Fedor Petrov.

If a polygon however exists, then there are several ways of making it unique, even in the convex case; you could ask either for minimal or maximal area or, you could demand its corners to be cocyclic, to name just some.