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?

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.