The answer to the second question is no as well.
Suppose $\dot\tau$ is a $Q$-name for a strategy for the player INC in the game $G_\kappa(P)$. Let us pretend that COM opens the game instead of INC and note that this is unproblematic. We will define by induction a descending sequence $\langle p_\alpha\mid\alpha<\kappa\rangle$ of conditions in $P$ and $Q$-names $\langle \dot p_\alpha\mid\alpha<\kappa\rangle$ so that
- $\Vdash_Q\check p_\beta\leq \dot p_\alpha\leq\check p_\alpha$ for all $\beta\leq\alpha$
- $\Vdash_Q$"The sequence $\langle \dot p_\beta\mid\beta\leq\alpha\rangle$ make up the fist $\alpha+1$-many moves of player COM in a game of $G_{\check\kappa}(\check P)$ in which INC plays according to $\dot\tau$ and the final response of $\dot \tau$ to this is a condition $\leq\check p_\alpha$."
for all $\alpha<\kappa$. It follows that if $G$ is $Q$-generic then COM wins against $\dot\tau^G$ by playing the conditions $\langle \dot p_\alpha^G\mid\alpha<\kappa\rangle$, so $\dot\tau$ is not a winning strategy for INC.
Let us turn to the construction. Assume $\alpha<\kappa$ and that $\dot p_\beta, p_\beta$ are defined for all $\beta<\alpha$. Let $s_0$ be some lower bound of $\langle p_\beta\mid\beta<\alpha\rangle$ which exists as $P$ is ${<}\kappa$-closed. Build descending sequences of maximal possible length
$$\langle s_\gamma\mid\gamma\leq\xi\rangle,\ \langle t_\gamma\mid\gamma<\xi\rangle$$
and an antichain $A_\xi:=\{q_\gamma\mid \gamma<\xi\}\subseteq Q$ so that
- $s_\gamma\leq t_\gamma\leq s_{\gamma+1}$ for all $\gamma<\xi$ and
- $q_\gamma\Vdash_Q$"if COM plays $\langle \dot p_\beta\mid\beta<\alpha\rangle^\frown\check s_\gamma$ and INC follows $\dot\tau$, INC's final play is $t_\gamma$".
The construction is straightforward and can only break down for two reasons: Either there is no lower bound of $\langle t_\gamma\mid\gamma<\xi\rangle$ or $A_\xi$ is a maximal antichain in $Q$. The latter must happen first since $P$ is ${<}\kappa$-closed and $Q$ is $\kappa$-cc. We set $p_\alpha$ to be the final $s_\xi$ and by mixing of names, we can find a $Q$-name $\dot p_\alpha$ so that
$$q_\gamma\Vdash_Q\dot p_\alpha=\check s_\gamma$$
for all $\gamma<\xi$. This completes the construction.
Finally, let me remark that this is sensitive to the order of play in $G_\kappa(P)$: If $G_\kappa'(P)$ is the game where INC plays at limit steps instead of COM then it is possible that INC has a winning strategy for $G_\kappa'(P)$ in $V^Q$:
Let $Q$ be the poset of finite partial functions $q:\omega_1\rightarrow 2$ ordered by inclusion and let $P=\mathrm{Add}(\omega_1, 1)$. $Q$ is ccc and $P$ is ${<}\omega_1$-closed. Let $G$ be $Q$-generic and consider the strategy for INC for the game $G_\kappa'(P)$ where INC adds a bit of information which agrees with the generic function $\bigcup G$ at the least ordinal which is not yet in the domain of any condition played so far. Assume toward a contradiction that some game in which INC follows this strategy lasts for $\omega_1$-many rounds. The result is a function $f\colon\omega_1\rightarrow 2$ which agrees with $\bigcup G$ on a club $C$. Now $Q$ is ccc, so there is a club $C'\subseteq C$ with $C'\in V$. It follows that $f\notin V$, but we must have $f\upharpoonright\alpha\in V$ for all $\alpha<\omega_1$. This is impossible by a result of Spencer Unger: As $Q\times Q$ is ccc, $Q$ has the $\omega_1$-approximation property, see Lemma 1.2 in Fragility and indestructibility II, Annals of Pure and Applied Logic 166 (2015) 1110-1122.
