Generating Uncountable Transformation Semigroups

We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions. (i) Is every countable subset F of S also a subset of a finitely generated subsemigroup of S? If so, what is the least number n such that for every countable subset F of S there exist n elements of S that generate a subsemigroup of S containing F as a subset. (ii) Given a subset U of S, what is the least cardinality of a subset A of S such that the union of A and U is a generating set for S? (iii) Define a preorder relation 4 on the subsets of S as follows. For subsets V and W of S write V 4 W if there exists a countable subset C of S such that V is contained in the semigroup generated by the union of W and C. Given a subset U of S, where does U lie in the preorder 4 on subsets of S? Semigroups S for which we answer question (i) include: the semigroups of the injective functions and the surjective functions on a countably infinite set; the semigroups of the increasing functions, the Lebesgue measurable functions, and the differentiable functions on the closed unit interval [0, 1]; and the endomorphism semigroup of the random graph.