MSOL optimization problems on graphs of bounded cliquewidth, with cardinality predicates
After asking around some more, it seems that the answers to 1) and 2) are both YES. Optimizing the cardinality of a set is possible in LinEMSOL (as mentioned by Martin Lackner); as I have been told, the existence of the cardinality predicates is no problem since they can be efficiently handled by finite-state tree automatons, which should follow (more explicitly than in the originally referenced paper) from On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width.