Climbing up subsets of $\omega_1$ using reals
"Every set is really climbable": this contradicts AC. By AC construct a set $X\subseteq \omega_1$ such that $L[X]\models $"$\omega_1 =\omega_1^V$". Then for arbitrarily large $\gamma < \omega_1$ we have that $L_\gamma[X\cap \gamma] \models $"Every set is countable". But such a set cannot be climbable, for if $f\in L[r]$ were a 'climbing' function, according to the definition we should have such $X\cap \gamma \in dom(f) \subset L[r]$ contradicting the inaccessibility of $\omega^V_1$ there.
On the other hand, assuming $AD$, for example in $L({R})$ assuming large cardinals (where also DC holds - but that is not relevant), by an early result of Solovay, every subset of $\omega_1$ is in some $L[r]$ for a real $r$; clearly then every such subset is really, and trivially, climbable.
Concerning Copy(K)-determinacy: you mention that every set of countable ordinals being climbable would lend plausibility to this in ZFC (that is with AC). We just saw that climbability fails under AC; of course $\forall K$Copy(K)-determinacy holds under AD; and if you make the game harder for II by insisting (I make it that II has the winning strategy, not I, by Boundedness) that II's wellorders code all of $K$ below their supremum, then one can conclude that $K$ is in $L[\tau]$ where $\tau$ is II's strategy. So for this slightly more rigorous game, its determinacy for all $K$ is inconsistent with choice and $\omega_1$ inaccessible to reals.