Given a non-measurable set and a cardinal does there exist a subset of that cardinality which is measurable?

Question

Let $A\subseteq\mathbb{R}$ not be Lebesgue-measurable. My questions then are:

1. If $\kappa\leq 2^{\aleph_0}$ is any cardinal such that $\kappa>\aleph_0$ does there then exist a measurable subset $M$ of $A$ such that $\lvert M\rvert=\kappa$?
2. If the answer for (1) is negative: Since, $\aleph_1\leq2^{\aleph_0}$ how would the answer in (1) change if we only required $\kappa\leq \aleph_1$ (obviously still assuming $\kappa>\aleph_0$).
3. Where applicable, does assuming the Continuum Hypothesis affect any of the previous two questions?