A congruence lattice of a universal algebra is a complete (therefore bounded) lattice, and is also algebraic (every element is a join of compact elements). As such, the collection C of congruence lattices of algebras of some class is a variety only when C is the collection of one element lattices.
The ' operation above considers the lattice variety L generated by C. This variety L is either trivial, or it contains the variety generated by the two element lattice. This last variety is the variety D of distributive lattices. If the beginning variety of algebras has a Mal'cev term for distributivity, then C and thus L consists of distributive lattices. Lattices have such a term: take the join of the three terms (x_i meet x_{i+1}) where the subscripts are taken modulo 3. Thus for any lattice variety L, L' is either D or trivial, as D is a minimal variety in the collection (lattice!) of lattice varieties.
Finally, for any starting variety V of algebras, we have V'=L for some lattice variety L. If L is trivial, call this T, then T'=T. Otherwise L'=D=D'. So the ' operation reaches one of two fixedpoints D or T within two steps or one. This shows that L contains L' for all lattice varieties L, with strict containment precisely when L has a nondistributive lattice. This answers Question 1. Question 2 is answered negatively: there are many two element descendingnon-ascending chains, all but one ending in D.