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For question 2, you can take a non-simple space-filling curve in one lower dimension $f:\mathbb{R} \twoheadrightarrow \mathbb{R}^{n-1}$ and follow it with a surjection $g:\mathbb{R}^{n-1} \twoheadrightarrow S^{n-1}$ where $S^{n-1}$ is the unit $(n-1)$-sphere in $\mathbb{R}^n$. For example, $g$ could be the exponential map on the tangent space on one point. Then the image of $t \mapsto e^t g(f(t))$ intersects every ray through the origin, and scaling by $e^t$ makes the curve simple.