$\Bbb R\subseteq\Bbb C$, so if $f$ is not injective on $\Bbb R$, it can’t be injective on $\Bbb C$, either. That is, if there are two different real numbers $x$ and $y$ such that $f(x)=f(y)$, those two real numbers are also complex numbers showing that $f$ is not injective on $\Bbb C$. Thus, for (2) all that’s left is to decide whether $f:\Bbb C\to\Bbb C$ is surjective. In other words, if $z\in\Bbb C$, is there always a complex number $w$ such that $w^2=z$? This is most easily answered if you know the exponential representation of complex numbers: every complex number can be written in the form $re^{i\theta}$, where $r,\theta\in\Bbb R$ and $r\ge 0$. Given such a number $re^{i\theta}$, can you find real numbers $s$ and $\varphi$ such that $\left(se^{i\varphi}\right)^2=re^{i\theta}$?
In (4) the domain of $f$ is the two coordinate axes in the complex plane. Alternatively, it’s the set of all $x+iy$ such that at least one of $x$ and $y$ is $0$, so it’s the set of all real numbers together with all purely imaginary numbers, i.e., numbers of the form $yi$. You already know that $f$ is not injective on the domain $\Bbb R$, which is included in this set, so you can conclude right away that $f$ is not injective on this set either. Thus, once again it comes down to deciding whether $f$ is surjective. Suppose that $r\in\Bbb R$. If $r\ge 0$ there is certainly an $x$ in the domain of $f$ such that $x^2=r$. What if $r<0$? For instance, is there a number of the form $yi$ whose square is $-4$?
Try (2) and (4), and if you get them, see whether the experience helps you with (5); if not, let me know, and I’ll add some hints for (5).
Added: For (5) let $D=\{x+iy:y>0\}\cup\{x:x\ge 0\}$, and consider the function $f:D\to\Bbb C:z\mapsto z^2\;.$ Pictorially speaking, $D$ is everything above the real axis in the complex plane together with all of the non-negative real numbers. In terms of the exponential representation of complex numbers, $D=\left\{re^{i\theta}:r\ge 0\text{ and }0\le\theta<\pi\right\}\;.$
Suppose that $se^{i\varphi}$ is any complex number. Show that you can always choose $s$ and $\varphi$ so that $s\ge0$ and $0\le\varphi<2\pi$, just as you can when you choose the polar coordinates of a point in the plane. Then find an $re^{i\theta}\in D$ such that $\left(re^{i\theta}\right)^2=se^{i\varphi}$; this will show that $f$ is surjective.
To decide whether $f$ is injective, suppose that $re^{i\theta},se^{i\varphi}\in D$ and $\left(re^{i\theta}\right)^2=\left(se^{i\varphi}\right)^2$, i.e., that $r^2e^{2i\theta}=s^2e^{2i\varphi}$. Knowing that $r,s\ge 0$ and $0\le\theta,\varphi<\pi$, can you show from this that $r=s$ and $\theta=\varphi$? If so, you’ll have proved that $f$ is injective.