Can manifold subsets always be made into submanifolds?

Your statements in Question A are all essentially correct. Whether or not you can put a manifold topology on a set depends only on the cardinality of the set. For sets with cardinality greater than that of $\mathbb R,$ the answer depends on your axioms:

• If your definition of manifold includes second-countability, then any set strictly larger than $\mathbb R$ cannot be made a manifold.
• If not, every set can be made a 0-dimensional manifold. (If you insist on positive dimension, I think by assuming the axiom of choice you can prove that all sets with cardinality at least that of $\mathbb R$ can be made manifolds.)

Regardless, I think this point is a tangent from your main concern:

Question B. You can answer this without needing to think about smooth structure at all: If $A$ has an atlas making it a smooth submanifold of $B,$ then $A$ is a topological submanifold of $B.$ This makes things much easier, since there is no arbitrary choice to be made when talking about topological submanifolds: a subset $A \subset B$ is a topological submanifold if and only the induced topology makes $A$ a topological manifold.

So, if $A$ has some atlas making it a smooth submanifold, then $A$ is a topological manifold in the induced topology. Thus any subset $A \subset B$ which is not a topological manifold in the induced topology (e.g. one that is not locally Euclidean) provides a counterexample.