Homeomorphism proof 
$\mathbb{R}^2-(0 \times \mathbb{R}_+) \approx \mathbb{R}^2$

Now consider the map that sends the line $(-1 \times \mathbb{R}_+)$ to $(0 \times \mathbb{R}_+)$. And then continue this inductively. Every function is a map, because it is a translation. The composition of any finite number of maps is a map. The question I have is whether the countably infinite composition of these maps is a map. Thank you.

Your idea is not bad, and it certainly works as a bijection between the given sets.
But the defined map is not going to be continuous.
Hint: Use polar coordinates, the angle measured starting from the given ray $\{0\}\times\Bbb R^+$ (i.e. the $y$-axis), then deform the angles from $(0,360^\circ)$ to $(90^\circ,270^\circ)$, thus you get the open half plane.
