Regarding the question about the strength of the ultrafilter lemma for distributive lattices, let me cite the relevant theorem from Herrlich's book:
Theorem 4.32. Equivalent are:
- Every lattice has a maximal filter.
- Every complete lattice has a maximal filter.
- Every distributive lattice has a maximal filter.
- Every closed lattice has a maximal filter.
- AC.
In this theorem a closed lattice is a lattice that is isomorphic to the lattice of closed subsets of a nonempty topological space (Definition 4.28).
Herrlich, Horst
Axiom of choice
Lecture Notes in Mathematics, 1876.
Springer-Verlag, Berlin, 2006.
Herrlich credits the equivalence of Item 3 and Item 5 to
G. Klimowsky.
El Theorema de Zorn y la existencia de filtros e ideales
maximales en los reticulados distributivos.
Rev. Union Mat. Argentina, 18:160-164, 1958.
Where Herrlich writes in terms of AC and in terms of the existence of a maximal filter, Klimowsky writes that Zorn's Lemma is equivalent to En todo reticulado distributivo con primer elemento, todo filtro está contenido en un ultrafiltro.