A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable

Assuming $AD$, a version of the continuum hypothesis holds: every set of reals is either countable or of size continuum (this is a consequence of the perfect set property). So assuming $AD$, your $A_{<2^{\aleph_0}}$ and Freiling's $AS$ are equivalent. In particular, both can hold in Solovay's model, so the answer to your question is "no."

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Your second question seems much more interesting. Off the top of my head, I don't see a reason why we can't have a model of ZF+DC+LM+$\neg$WCH (except of course for the small fact that I don't know how to build one).