for question 1) it's not clear to me why $C^2$ approximating $z \mapsto z^k$ by immersions is possible at all. do you have an example of such a sequence?

edit: Ok, I thought some more on the topological issue of whether it's always possible to deform $z \mapsto z^k$ to an immersion in $C^2$ and I can say that it is definitely NOT possible  if $k$ is even. In particular, it's impossible when $k=2$. There is an easy necessary condition for an existence of such deformation. An immersion $f$ of a disk gives a parallelization of the tangent bundle along $f$, i.e we get a map $f'\colon D^2\to V_2(\mathbb R^3)$ where $V_2(\mathbb R^3)$ is the Stiefel manifold of  orthonormal  2-frames in $\mathbb R^3$ which is of course just $SO(3)$. This map ought to extend the map on the boundary of the disk $S^1$ which being an immersion already won't change much under a small deformation. That map is essentially given by $z\to kz^{k-1}$ which disregarding the conformal factor $k$ can be thought of as a map $S^1\to SO(2)=V_2^{or}(\mathbb R^2)\subset V_2(\mathbb R^2)=0(2)$. The natural map $V_2(\mathbb R^2)\to V_2(\mathbb R^3)$  corresponds to the standard inclusion $SO(2)\to SO(3)$ on identity component. Now, $\pi_1(SO(2))\cong \mathbb Z$ and $\pi_1(SO(3))\cong \mathbb Z/2$ and the map $\pi_1(SO(2))\to \pi_1(SO(3))$ is well-known to be onto (as is immediate from the long exact sequence for the fibration $SO(2)\to SO(3)\to S^2$).

 That means that the map $z\mapsto z^{k-1}$ gives a generator of $\pi_1(SO(3))$ when $k$ is even  and thus can not be extended to $D^2$.

I think the above necessary condition should also be sufficient  and therefore when $k$ is odd  such a deformation should always possible (at least in $C^0$) as evidenced by $k=1$ of course. I'm not at all an expert on immersions but the subject is very well understood and I hope someone who knows more about it will chime in.



For question 2) you can not expect any upper curvature bounds. Given an  immersion you can locally  perturb it on an arbitrary small neighborhood of 0 by adding a small thin "finger" to your surface. this will introduce some arbitrary positive (and negative) curvature. Doing this along a given sequence converging in $C^2_{loc}(\mathbb D\backslash \{0\})\cap C^0(\mathbb D)$ will keep such convergence.