3) Realization takes $S^{1,1}$ to $S^{1}$ and $S^{0,0}$ to $S^0$, so you just have to show the realization is the non-zero element of $\pi_1(S^0)$. But the cofiber of the motivic Hopf map is $\Sigma^{-2,-1} \Sigma^\infty \mathbb{P}^2$, which realizes to $\Sigma^{-2} \mathbb{C}P^2$, I think.

Here is a first thought: 1) This is true; I don't actually know who this is first due to, but it must be in Toda's book "Composition Methods in Homotopy Groups of Spheres". 2) Do you mean complex oriented cohomology theories? For such an $E$, the Hurewicz map $\pi_*S \to \pi_*E$ factors through the torsion free ring $\pi_*MU$, so this should follow.

If so, one would expect it to arise from en.wikipedia.org/wiki/Five-term_exact_sequence, but the indicies don't quite line up

Maybe Fausk's work will help? sciencedirect.com/science/article/pii/S0022404902001457

The answer to your question seems to be no: link.springer.com/chapter/10.1007%2FBFb0069236#page-1

For the last question, see mathoverflow.net/questions/67227/…