Melzak, Aberth and Beyond

Melzak's problem: Among all polyhedrons with edge length sum = 1, which one has the max volume?

Aberth's problem: Among all polyhedrons with edge length sum = 1, which one has max total surface area?

Both these questions are not fully answered. For a detailed survey there is the PhD thesis of Scott Berger (2001). The conjectured answer to Melzak is a right triangular prisms. For Aberth, the guess is that there is no such well defined polyhedron (in the same sense that there is no unique smallest positive real number).

I couldn't find references on these variants of the above questions:

1. With fixed edge length sum AND surface area, which polyhedron maximizes the volume?

2. With fixed edge length sum AND volume, which polyhedron maximizes surface area?

Note: With given area and perimeter, there are in general, infinitely many triangles. This implies for a given edge length sum and surface area, there can be infinitely many right triangular prisms with the same volume. Does this mark a useful starting point in answering question 1?