I have two, somehow related questions, and I would be very grateful if you point out at some references if the answer is known. If these are too elementary or not research level, then please feel free to move it mathexchange. I am not an expert in this, so I thought it is worth asking here.

According to Ravenel's orange book, Nilpotence Theorem implies that $H\mathbb{F}$ and $K(n)$ are ``essentially'' the only homology theories for which Kunneth isomorphism holds; I suppose this latter means that $$E_*(X\times Y)\simeq E_*(X)\otimes_{E_*(pt)}E_*(Y).$$ So, my first question is that if this conclusion is immediate or it is not that trivial and a proof is recorded somewhere? I have not tried to prove it.

Second, I wonder if there is a spectrum $E$ so that $$\pi_*L_E(S^0\wedge S^0)\simeq\pi_*L_ES^0\otimes\pi_*L_ES^0$$ where the tensor product is over some suitable coefficient ring, $L_E$ is the Bousfield localisation with respect to $E$, and $S^0$ is the sphere spectrum. Or, equivalently on the category of spaces $$\pi_*L_E(X\times Y)\simeq\pi_*L_EX\otimes\pi_*L_EY$$ again with tensor product over say $\pi_*L_ES^0$. A first guess is $E=H\mathbb{F}$ and $K(n)$, but these do not seem that immediate to me either.